Probability
152 JEE Maths previous year questions on Probability — options free on every question; 15 include the answer & explanation free, the rest unlock with PYQ Pass.
Consider an event \(E\) such that a matrix of order \(2 \times 2\) is invertible with entries 0 or 1. Then, \(P(E)\) is (where \(P(X)\) denotes the probability of event \(X\) )
\(\frac{3}{8}\)
\(\begin{aligned}& 2 \times 2 \rightarrow \text { Matrix } \\& |A| \neq 0 \\& n(s)=2 \times 2 \times 2 \times 2 \Rightarrow 16\end{aligned}\)
\(A\) can't be
\(\begin{aligned}& {\left[\begin{array}{ll}0 & 0 \\0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 1 \\1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\0 & 0\end{array}\right]} \\& {\left[\begin{array}{ll}1 & 1 \\0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\1 & 1\end{array}\right]} \\ & {\left[\begin{array}{ll}0 & 1 \\0 & 1\end{array}\right],\left[\begin{array}{ll}0 & 0 \\1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\0 & 1\end{array}\right]} \end{aligned}\)
\(\left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right]\)
So, total \(\Rightarrow 10\) types of matrix we can't take
\(\begin{aligned}& \therefore P(\bar{A})=\frac{10}{16}=\frac{5}{8} \\& \therefore P(A)=1-P(\bar{A})=1-\frac{5}{8} \Rightarrow \frac{3}{8} \\\end{aligned}\)
If the probability that the random variable X takes the value x is given by \(\mathrm{P}(\mathrm{X}=\mathrm{x})=\mathrm{k}(\mathrm{x}+1){3}^{-\mathrm{x}}\), \(\mathrm{x}=0,1,2,3\ldots ..\), where k is a constant, then \(P\left(X\geq 3\right)\) is equal to
[JEE Main 2025, 3 Apr (Shift 2)]
\(\frac{1}{9}\)
We have
\(P(X=x)=k(x+1){3}^{−x},\ x=0,1,2,\ldots\)
Use \({\sum }_{x=0}^{\mathrm{∞}}(x+1){r}^{x}=\frac{1}{(1−r{)}^{2}}\) (for \(\mathrm{∣}r\mathrm{∣}<1\)) with \(r=\frac{1}{3}\):
\(\sum _{x=0}^{\mathrm{∞}}(x+1){(\frac{1}{3})}^{x}=\frac{1}{{(1−\frac{1}{3})}^{2}}=\frac{1}{{(\frac{2}{3})}^{2}}=\frac{9}{4}\)
Since total probability is 1:
\(k⋅\frac{9}{4}=1\Rightarrow k=\frac{4}{9}\)
Now
\(P(X\leq 2)=\sum _{x=0}^{2}k(x+1){3}^{−x}=\frac{4}{9}[1+\frac{2}{3}+\frac{3}{9}]=\frac{4}{9}⋅2=\frac{8}{9}\)
So
\(P(X\geq 3)=1−P(X\leq 2)=1−\frac{8}{9}=\frac{1}{9}\mathrm{.}\)
Two balls are selected at random one by one without replacement from a bag containing \(4\) white and \(6\) black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is \(\frac{\mathrm{m}}{\mathrm{n}},\) Where \(\gcd (\mathrm{m},\mathrm{n})=1\text{,}\) then \(\mathrm{m}+\mathrm{n}\) is equal to
[JEE Main 2025, 22 Jan (Shift 1)]
\(14\)
The probability that the first ball is black is \(\frac{6}{10}\)
After selecting a black ball, there are \(5\) black balls
left out of \(9\) total balls.
So, the probability that the second ball is black is \(\frac{5}{9}\)
\(P(A\cap B)=\frac{6}{10}\cdot \frac{5}{9}=\frac{30}{90}=\frac{1}{3}\)
Now, \(P(B)=\frac{6}{10}\times \frac{5}{9}+\frac{4}{10}\times \frac{6}{9}=\frac{1}{3}+\frac{4}{15}=\frac{5}{15}+\frac{4}{15}\)\(=\frac{9}{15}=\frac{3}{5}\)
By conditional probability,
\(P\left(A∣B\right)=\frac{P(A\cap B)}{P(B)}=\frac{\frac{1}{3}}{\frac{3}{5}}=\frac{1}{3}\cdot \frac{5}{3}\)\(=\frac{5}{9}\)
The probability \(\text{ }P(A∣B)=\frac{5}{9}\text{.}\)
\(\text{Here, }m=5\text{ and }n=9\text{, and }\gcd (5,9)=1\text{. }\)
\(m+n=5+9=14\)
Let \(A=\left[a_{i j}\right]\) be a \(2 \times 2\) matrix such that \(a_{i j} \in\{0,1\}\) for all \(i\) and \(j\). Let the random variable \(X\) denote the possible values of the determinant of the matrix A . Then, the variance of X is :
\(\frac{3}{8}\)
\(\left|A\right|=\left|\begin{matrix}{a}_{11} & {a}_{12} \\ {a}_{21} & {a}_{22}\end{matrix}\right|\)
\(={a}_{11}{a}_{22}-{a}_{21}{a}_{12}\\ \text{possible values for X}={-1,0,1}\)
\(\begin{matrix}\mathrm{x} & {\mathrm{P}}_{\mathrm{i}} & {\mathrm{P}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}} & {\mathrm{P}}_{\mathrm{i}}{{\mathrm{X}}_{\mathrm{i}}}^{2} \\ -1 & \frac{3}{16} & -\frac{3}{16} & \frac{3}{16} \\ 0 & \frac{10}{16} & 0 & 0 \\ 1 & \frac{3}{16} & \frac{3}{16} & \frac{3}{16} \\ & & \sum {\mathrm{P}}_{\mathrm{i}}{X}_{\mathrm{i}}=0 & \sum {\mathrm{P}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}^{2}=\frac{3}{8}\end{matrix}\)
\(∴var(\mathrm{x})=\sum {\mathrm{P}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}^{2}-{\left(\sum {\mathrm{P}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}\right)}^{2}\\ =\frac{3}{8}-0=\frac{3}{8}\)
A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope, just two consecutive letters AN are visible.The probability that the letter came from ANANTPUR is:
[JEE Main 2026, 5 Apr (Shift 1)]
\(\frac{10}{17}\)
Selection of any \(1\) out of \(2\) words \(=\frac{1}{2}\)
ANANTPURTgQPHd|[] (\(9\) letters):
Consecutive pairs are (A,N), (N,A), (A,N), (N,T), (T,P), (P,U), (U,R).
Total = \(7\) and "AN" appears \(2\) times.
P(AN|Anantpur) \(=\frac{2}{7}\)
KANPURTgQPHd|[] (\(6\) letters):
Consecutive pairs are (K,A), (A,N), (N,P), (P,U), (U,R).
Total = \(5\) and "AN" appears \(1\) time.
P(AN|Kanpur) \(=\frac{1}{5}\)
\(\therefore\) Required Probability \(=\frac{\frac{1}{2} \times \frac{2}{7}}{\frac{1}{2} \times \frac{1}{5}+\frac{1}{2} \times \frac{2}{7}}=\frac{\frac{2}{7} \times 5}{\frac{17}{7}}=\frac{10}{17}\)
One die has two faces marked \(1\), two faces marked \(2\), one face marked \(3\) and one face marked \(4\). Another die has one face marked \(1,\) two faces marked \(2\), two faces marked \(3\) and one face marked \(4\). The probability of getting the sum of numbers to be \(4\) or \(5\), when both the dice are thrown together, is
[JEE Main 2025, 23 Jan (Shift 1)]
\(\frac{1}{2}\)
outcomes for one dice =\(\left\{1,1,2,2,3,4\right\}\)
outcomes for another dice =\(\left\{1,2,2,3,3,4\right\}\)
Identifying Favorable Outcomes for a Sum of \(4\):
\( (1, 3) \): Probability = \(\frac{1}{3}\times \frac{1}{3}=\frac{1}{9}\)
\( (2, 2) \): Probability \(=\frac{1}{3}\times \frac{1}{3}=\frac{1}{9}\)
\( (3, 1) \): Probability \(=\frac{1}{6}\times \frac{1}{6}=\frac{1}{36}\)
Identifying Favorable Outcomes for a Sum of \(5\):
\( (1, 4) \): Probability \(=\frac{1}{3}\times \frac{1}{6}=\frac{1}{18}\)
\( (2, 3) \): Probability \(=\frac{1}{3}\times \frac{1}{3}=\frac{1}{9}\)
\( (3, 2) \): Probability \(=\frac{1}{6}\times \frac{1}{3}=\frac{1}{18}\)
\( (4, 1) \): Probability \(=\frac{1}{6}\times \frac{1}{6}=\frac{1}{36}\)
Calculating the Total Probability:
\(P(\text{Sum}=4\text{or}\text{Sum}=5)=P(\text{Sum}=4)+P(\text{Sum}=5)\)
\(=\left(\frac{1}{9}+\frac{1}{9}+\frac{1}{36}\right)+\left(\frac{1}{18}+\frac{1}{9}+\frac{1}{18}+\frac{1}{36}\right)\)
\(=\frac{1}{2}\)
A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is
[JEE Main 2024, 29 Jan (Shift 1)]
\(\frac{5}{11}\)
Probability of getting \(2\) = \(\frac{1}{6}\)
Probability of getting other than \(2\) = \(\frac{5}{6}\)
Required Probability \(=\frac{5}{6}\times \frac{1}{6}+\frac{5}{6}\times \frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}+\ldots \ldots \ldots\)
\(=\frac{5}{36}\left[1+{\left(\frac{5}{6}\right)}^{2}+{\left(\frac{5}{6}\right)}^{4}+\ldots \ldots \ldots \right]=\frac{5}{36}\times \frac{1}{1−\frac{25}{36}}=\frac{5}{11}\)
A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is
[JEE Main 2024, 29 Jan (Shift 1)]
\(\frac{5}{11}\)
Probability of getting \(2\) = \(\frac{1}{6}\)
Probability of getting other than \(2\) = \(\frac{5}{6}\)
Required Probability \(=\frac{5}{6}\times \frac{1}{6}+\frac{5}{6}\times \frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}+\ldots \ldots \ldots\)
\(=\frac{5}{36}\left[1+{\left(\frac{5}{6}\right)}^{2}+{\left(\frac{5}{6}\right)}^{4}+\ldots \ldots \ldots \right]=\frac{5}{36}\times \frac{1}{1−\frac{25}{36}}=\frac{5}{11}\)
If the probability that the random variable X takes the value x is given by \(\mathrm{P}(\mathrm{X}=\mathrm{x})=\mathrm{k}(\mathrm{x}+1){3}^{-\mathrm{x}}\), \(\mathrm{x}=0,1,2,3\ldots ..\), where k is a constant, then \(P\left(X\geq 3\right)\) is equal to
[JEE Main 2025, 3 Apr (Shift 2)]
\(\frac{1}{9}\)
We have
\(P(X=x)=k(x+1){3}^{−x},\ x=0,1,2,\ldots\)
Use \({\sum }_{x=0}^{\mathrm{∞}}(x+1){r}^{x}=\frac{1}{(1−r{)}^{2}}\) (for \(\mathrm{∣}r\mathrm{∣}<1\)) with \(r=\frac{1}{3}\):
\(\sum _{x=0}^{\mathrm{∞}}(x+1){(\frac{1}{3})}^{x}=\frac{1}{{(1−\frac{1}{3})}^{2}}=\frac{1}{{(\frac{2}{3})}^{2}}=\frac{9}{4}\)
Since total probability is 1:
\(k⋅\frac{9}{4}=1\Rightarrow k=\frac{4}{9}\)
Now
\(P(X\leq 2)=\sum _{x=0}^{2}k(x+1){3}^{−x}\)\(=\frac{4}{9}[1+\frac{2}{3}+\frac{3}{9}]=\frac{4}{9}⋅2=\frac{8}{9}\)
So
\(P(X\geq 3)=1−P(X\leq 2)=1−\frac{8}{9}=\frac{1}{9}\mathrm{.}\)
A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability, that A wins if A makes the first throw, is
[JEE Main 2025, 24 Jan (Shift 1)]
\(\frac{9}{19}\)
\(P(A)=\frac{4}{36}=\frac{1}{9},P(B)=\frac{5}{36}\\ P(\text{ A wins })=\frac{4}{36}+\frac{32}{36}\cdot \frac{31}{36}\cdot \frac{4}{36}+\ldots ...\infty \\ P(\text{ A wins })=\frac{\frac{4}{36}}{1-\frac{32}{36}\cdot \frac{31}{36}}\\ =\frac{\frac{1}{9}}{1-\frac{8}{9}\cdot \frac{31}{36}}\\ =\frac{\frac{1}{9}}{1-\frac{62}{81}}=\frac{9}{19}\)
Let the sum of two positive integers be \(24.\) If the probability, that their product is not less than \(\frac{3}{4}\) times their greatest possible product, is \(\frac{m}{n}\), where \(\operatorname{gcd}(m, n)=1\), then \(n-m\) equals
[JEE Main 2024, 8 Apr (Shift 1)]
10
\( \mathrm{x}+\mathrm{y}=24, \mathrm{x}, \mathrm{y} \in \mathrm{~N} \)
\( \mathrm{AM}>\mathrm{GM} \Rightarrow \mathrm{xy} \leq 144 \)
\( x y \geq 108\)
Favorable pairs of \((x, y)\) are
\({(13,11),(12,12),(14,10),(15,9),(16,8),(17,7),\\ (18,6),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13)}\)
i.e. \(13\) cases
Total choices for \(\mathrm{x}+\mathrm{y}=24\) is \(23\)
Probability \(\frac{13}{23}=\frac{m}{n}\)
\(n-m=10\)
Let the sum of two positive integers be \(24.\) If the probability, that their product is not less than \(\frac{3}{4}\) times their greatest possible product, is \(\frac{m}{n}\), where \(\operatorname{gcd}(m, n)=1\), then \(n-m\) equals
[JEE Main 2024, 8 Apr (Shift 1)]
10
\( \mathrm{x}+\mathrm{y}=24, \mathrm{x}, \mathrm{y} \in \mathrm{~N} \)
\( \mathrm{AM}>\mathrm{GM} \Rightarrow \mathrm{xy} \leq 144 \)
\( x y \geq 108\)
Favorable pairs of \((x, y)\) are
\({(13,11),(12,12),(14,10),(15,9),(16,8),(17,7),\\ (18,6),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13)}\)
i.e. \(13\) cases
Total choices for \(\mathrm{x}+\mathrm{y}=24\) is \(23\)
Probability \(\frac{13}{23}=\frac{m}{n}\)
\(n-m=10\)
A bag contains \(19\) unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is \(\frac{\mathrm{m}}{\mathrm{n}},\gcd (\mathrm{m},\mathrm{n})=1\), then \({n}^{2}-{m}^{2}\)is equal to :
[JEE Main 2025, 7 Apr (Shift 2)]
\(80\)
- \(U\): Selecting an unbiased coin. \(P(U)=\frac{19}{20}\)
- \(B\): Selecting the biased (two-headed) coin. \(P(B)=\frac{1}{20}\)
- \(H\): Getting a head on the toss.
- \(P(H\mathrm{∣}U)=\frac{1}{2}\) (for an unbiased coin)
- \(P(H\mathrm{∣}B)=1\) (for a two-headed coin)
\(P(U\mathrm{∣}H)=\frac{P(U)⋅P(H\mathrm{∣}U)}{P(U)⋅P(H\mathrm{∣}U)+P(B)⋅P(H\mathrm{∣}B)}\)
\(P(U\mathrm{∣}H)=\frac{\frac{19}{20}⋅\frac{1}{2}}{(\frac{19}{20}⋅\frac{1}{2})+(\frac{1}{20}⋅1)}=\frac{\frac{19}{40}}{\frac{19}{40}+\frac{2}{40}}=\frac{19}{21}\)
- Given \(P(U\mathrm{∣}H)=\frac{m}{n}=\frac{19}{21}\), where \(\text{gcd}(19,21)=1\).
- So, \(m=19\) and \(n=21\).
\({n}^{2}−{m}^{2}=(n−m)(n+m)\)
\({n}^{2}−{m}^{2}=(21−19)(21+19)=2\times 40=80\)
If A and B are two events such that \(P(A)=0.7\),\(\mathrm{P}(\mathrm{B})=0.4\) and \(\mathrm{P}(\mathrm{A}\cap \overset{¯}{\mathrm{B}})=0.5\), where \(\overset{¯}{\mathrm{B}}\) denotes the complement of B, then \(P(B∣(A\cup \overset{¯}{B}))\) is equal:-
[JEE Main 2025, 8 Apr (Shift 1)]
\(\frac{1}{4}\)
\(P(A)=0.7\) \(P(B)=0.4\) \(P(A\cap \overset{‾}{B})=0.5\)
\(\begin{matrix}P(A)−P(A\cap B)=0.5 \\ P(A\cap B)=0.2\end{matrix}\)
\(P(\frac{B}{A\cup \overset{-}{B}})=\frac{P(B\cap (A\cup \overset{-}{B}))}{P(A\cup \overset{-}{B})}=\frac{0.2}{0.8}=\frac{1}{4}\)
WHERE
\(\begin{matrix}P(A\cup \overset{‾}{B})=P(A)+P(\overset{‾}{B})−P(A\cap \overset{‾}{B}) \\ =0.7+0.6−0.5 \\ =0.8\end{matrix}\)
A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability, that A wins if A makes the first throw, is
[JEE Main 2025, 24 Jan (Shift 1)]
\(\frac{9}{19}\)
\(P(A)=\frac{4}{36}=\frac{1}{9},P(B)=\frac{5}{36}\\ P(\text{ A wins })=\frac{4}{36}+\frac{32}{36}\cdot \frac{31}{36}\cdot \frac{4}{36}+\ldots ...\infty \\ P(\text{ A wins })=\frac{\frac{4}{36}}{1-\frac{32}{36}\cdot \frac{31}{36}}\\ =\frac{\frac{1}{9}}{1-\frac{8}{9}\cdot \frac{31}{36}}\\ =\frac{\frac{1}{9}}{1-\frac{62}{81}}=\frac{9}{19}\)
Two persons \(A\) and \(B\) throws a pair of dice alternatively. For \(A\) to win he should throw sum of 5 before \(B\) throws sum of 8 . If \(A\) throws first, then the probability that \(A\) wins, is (24 Jan, Shift I, Memory Based)
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Let \(A\) and \(B\) are two events such that \(P(A \cap B)=\frac{1}{10}\) and \(P(A / B)\) and \(P(B / A)\) are the roots of the equation \(12 x^2-7 x+1=0\), then \(\frac{P(\bar{A} \cup \bar{B})}{P(\bar{A} \cap \bar{B})}\) is equal to (22 Jan, Shift II, Memory Based)
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\(\text{ If }{k}_{1},{k}_{2}\in z\text{, then find probability of }{i}^{{k}_{1}}+{i}^{{k}_{2}}\text{ is non-zero }\) (28 Jan, Shift I, Memory Based)
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A 2×2 matrix form by elements {0,1} and random variable x be defined as value of determinat, then the variance is
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There are 3 bags \({B}_{1},{B}_{2},{B}_{3}\) which contain 4 white and 6 black, 6 white and 4 black, 5 white and 5 black balls respectively If 1 ball is randomly selected from a bag and is found to be white . Find the probability that ball came from bag \({B}_{2}\).
[JEE Main 2025]
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Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is
[JEE Main 2025, 28 Jan (Shift 1)]
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Two distinct numbers \(a\) and \(b\) are selected at random from \(1,2,3, \ldots, 50\). The probability, that their product \(a b\) is divisible by 3 , is
[JEE Main 2026, 22 Jan (Shift 1)]
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A die has 2 faces of 1, 2 faces of 3, 1 face of 2, 1 face of 4. Another die having 2 faces of 2, 2 faces of 1, 1 face of 3, 1 face of 4 are tossed. Find probability of getting sum 4 or 5.
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A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is \(\frac{1}{2}\). Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is \(\frac{1}{6}\). Then the probability that the student knows the answer of a randomly chosen question is
[JEE Advanced 2024]
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Two numbers \({\mathrm{k}}_{1}\text{ and }{\mathrm{k}}_{2}\) are randomly chosen from the set of natural numbers. Then, the probability that the value of \({\mathrm{i}}^{{\mathrm{k}}_{1}}+{\mathrm{i}}^{{\mathrm{k}}_{2}},(\mathrm{i}=\sqrt{-1})\) is non-zero, equals
[JEE Main 2025, 28 Jan (Shift 1)]
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The probability that certain electronic component fails when first used is 0.10 . If it does not fail immediately, the probability that it lasts for one year is 0.99 . The probability that a new component will last for one year is
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Let \(A\) and \(B\) are two events such that \(P(A \cap B)=\frac{1}{10}\) and \(P(A / B)\) and \(P(B / A)\) are the roots of the equation \(12 x^2-7 x+1=0\), then \(\frac{P(\bar{A} \cup \bar{B})}{P(\bar{A} \cap \bar{B})}\) is equal to (22 Jan, Shift II, Memory Based)
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Two numbers \({\mathrm{k}}_{1}\text{ and }{\mathrm{k}}_{2}\) are randomly chosen from the set of natural numbers. Then, the probability that the value of \({\mathrm{i}}^{{\mathrm{k}}_{1}}+{\mathrm{i}}^{{\mathrm{k}}_{2}},(\mathrm{i}=\sqrt{-1})\) is non-zero, equals
[JEE Main 2025, 28 Jan (Shift 1)]
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A bag contains 10 balls out of which \(k\) are red and \((10-k)\) are black, where \(0 \leq k \leq 10\). If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:
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Let \(A=\left[a_{i j}\right]\) be a square matrix of order \(2\) with entries either \(0\) or \(1\) . Let \(E\) be the event that \(A\) is an invertible matrix. Then the probability \(\mathrm{P}(\mathrm{E})\) is :
[JEE Main 2025, 24 Jan (Shift 2)]
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A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \( \mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64\left(\mu+\sigma^2\right)\) is
[JEE Main 2025, 22 Jan (Shift 1)]
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A bag contains 10 balls out of which \(k\) are red and \((10-k)\) are black, where \(0 \leq k \leq 10\). If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:
[JEE Main 2026, 28 Jan (Shift 1)]
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Bag \(1\) contains \(4\) white balls and \(5\) black balls, and Bag \(2\) contains n white balls and \(3\) black balls. One ball is drawn randomly from Bag \(1\) and transferred to Bag \(2\). A ball is then drawn randomly from Bag \(2\). If the probability, that the ball drawn is white, is \(\frac{29} { 45}\), then \(n\) is equal to :
[JEE Main 2025, 29 Jan (Shift 2)]
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A bag contains \((N+1)\) coins - \(N\) fair coins, and one coin with 'Head' on both sides. A coin is selected at random and tossed. If the probability of getting 'Head' is \(\frac{9}{16}\) then \(N\) is equal to:
[JEE Main 2026, 6 Apr (Shift 1)]
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A 2×2 matrix form by elements {0,1} and random variable x be defined as value of determinat, then the variance is
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A company has two plants \(A\) and \(B\) to manufacture motorcycles. \(60\)% motorcycles are manufactured at plant \(A\) and the remaining are manufactured at plant \(B\) .\(80\)% of the motorcycles manufactured at plant \(A\) are rated of the standard quality, while \(90\)% of the motorcycles manufactured at plant \(B\) are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If \(p\) is the probability that it was manufactured at plant \(B\) , then \(126 p\) is
[JEE Main 2024, 06 Apr (Shift 1)]
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A company has two plants \(A\) and \(B\) to manufacture motorcycles. \(60\)% motorcycles are manufactured at plant \(A\) and the remaining are manufactured at plant \(B\) .\(80\)% of the motorcycles manufactured at plant \(A\) are rated of the standard quality, while \(90\)% of the motorcycles manufactured at plant \(B\) are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If \(p\) is the probability that it was manufactured at plant \(B\) , then \(126 p\) is
[JEE Main 2024, 06 Apr (Shift 1)]
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Two distinct numbers \(a\) and \(b\) are selected at random from \(1,2,3, \ldots, 50\). The probability, that their product \(a b\) is divisible by 3 , is
[JEE Main 2026, 22 Jan (Shift 1)]
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A bag contains \(19\) unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is \(\frac{\mathrm{m}}{\mathrm{n}},\gcd (\mathrm{m},\mathrm{n})=1\), then \({n}^{2}-{m}^{2}\) is equal to :
[JEE Main 2025, 7 Apr (Shift 2)]
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A bag contains \(6\) blue and \(6\) green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is:
[JEE Main 2026, 6 Apr (Shift 2)]
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A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability, that A wins if A makes the first throw, is
[JEE Main 2025, 24 Jan (Shift 1)]
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Let a random variable X take values \(0,1,2,3\) with \(\mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)=\mathrm{p},\mathrm{P}(\mathrm{X}=2)=\mathrm{P}(\mathrm{X}=3)\) and \(E\left(X^2\right)=2 E(X)\). Then the value of \(8p-1\) is :
[JEE Main 2025, 7 Apr (Shift 2)]
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An integer is chosen at random from the integers \(1,2,3, \ldots, 50\). The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is
[JEE Main 2024, 29 Jan (Shift 2)]
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An integer is chosen at random from the integers \(1,2,3, \ldots, 50\). The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is
[JEE Main 2024, 29 Jan (Shift 2)]
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A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are \(\frac{2}{5},\frac{1}{5}\) and \(\frac{2}{5}\). The probabilities that the candidate reaches late at the examination centre are \(\frac{1}{5},\frac{1}{3}\) and \(\frac{1}{4}\), if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:
[JEE Main 2026, 8 Apr (Shift 2)]
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If A and B are independent events such that \(P\left(A\right)=\frac{1}{5},\text{ }P\left(A\cup B\right)=\frac{7}{10},\) then what is \(P\left(\overset{¯}{B}\right)\) equal to?
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Consider an event \(E\) such that a matrix of order \(2 \times 2\) is invertible with entries 0 or 1. Then, \(P(E)\) is (where \(P(X)\) denotes the probability of event \(X\) )
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One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is
[JEE Main 2025, 23 Jan (Shift 1)]
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Bag \(A\) contains \(3\) white, \(7\) red balls and Bag \(B\) contains \(3\) white, \(2\) red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is
[JEE Main 2024, 30 Jan (Shift 2)]
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Bag \(A\) contains \(3\) white, \(7\) red balls and Bag \(B\) contains \(3\) white, \(2\) red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is
[JEE Main 2024, 30 Jan (Shift 2)]
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If A and B are two events such that \(P(A \cap B)=0.1\), and \(P(A \mid B)\) and \(P(B \mid A)\) are the roots of the equation \(12 x^2-7 x+1=0\), then the value of \(\frac{\mathrm{P}(\overline{\mathrm{A}} \cup \overline{\mathrm{B}})}{\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})}\) is:
[JEE Main 2025, 22 Jan (Shift 2)]
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Let \(A=\left[a_{i j}\right]\) be a square matrix of order 2 with entries either 0 or 1 . Let \(E\) be the event that \(A\) is an invertible matrix. Then the probability \(\mathrm{P}(\mathrm{E})\) is :
[JEE Main 2025, 24 Jan (Shift 2)]
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\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of \(5\) before \(B\) throws a sum of \(8\), and \(B\) wins if he throws a sum of \(8\) before \(A\) throws a sum of \(5\). The probability, that \(A\) wins if \(A\) makes the first throw, is
[JEE Main 2025, 24 Jan (Shift 1)]
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The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is:
[JEE Main 2025, 4 Apr (Shift 1)]
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A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is \(\frac{1}{2}\). Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is \(\frac{1}{6}\). Then the probability that the student knows the answer of a randomly chosen question is
[JEE Advanced 2024]
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If A and B are two events such that \(P(A \cap B)=0.1\), and \(P(A \mid B)\) and \(P(B \mid A)\) are the roots of the equation \(12 x^2-7 x+1=0\), then the value of \(\frac{\mathrm{P}(\overline{\mathrm{A}} \cup \overline{\mathrm{B}})}{\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})}\) is:
[JEE Main 2025, 22 Jan (Shift 2)]
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A man throws a fair coin repeatedly. He gets \(10\) points for each head he throws and \(5\) points for each tail he throws. If the probability that he gets exactly \(30\) points is \(\frac{m}{n},\gcd (m,n)=1\), then \(m+n\) is equal to:
[JEE Main 2026, 2 Apr (Shift 2)]
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If the probability that the random variable \(X\) takes the value \(x\) is given by \(\mathrm{P}(\mathrm{X}=\mathrm{x})=\mathrm{k}(\mathrm{x}+1){3}^{-\mathrm{x}}\), \(\mathrm{x}=0,1,2,3\ldots ..\), where \(k\) is a constant, then \(P\left(X\geq 3\right)\) is equal to
[JEE Main 2025, 3 Apr (Shift 2)]
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If \(\ S \) is a set of words formed by all the letters of word "GARDEN", then find the probability that all vowels are not in alphabatical order:
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If A and B are two events such that \(P(A)=0.7\), \(\mathrm{P}(\mathrm{B})=0.4\) and \(\mathrm{P}(\mathrm{A}\cap \overset{¯}{\mathrm{B}})=0.5\), where \(\overset{¯}{\mathrm{B}}\) denotes the complement of B, then \(P(B∣(A\cup \overset{¯}{B}))\) is equal:-
[JEE Main 2025, 8 Apr (Shift 1)]
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Bag \(B_1\) contains 6 white and 4 blue balls, Bag \(B_2\) contains 4 white and 6 blue balls, and Bag \(B_3\) contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag \(B_2\), is :
[JEE Main 2025, 28 Jan (Shift 2)]
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A die has 2 faces of 1, 2 faces of 3, 1 face of 2, 1 face of 4. Another die having 2 faces of 2, 2 faces of 1, 1 face of 3, 1 face of 4 are tossed. Find probability of getting sum 4 or 5.
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The coefficients \(a,b,c\)in the quadratic equation \(a{x}^{2}+bx+c=0\) are from the set \(\left\{1,2,3,4,5,6\right\}\). If the probability of this equation having one real root bigger than the other is \(p\), then \(216p\) equals :
[JEE Main 2024, 5 Apr (Shift 2)]
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The coefficients \(a,b,c\)in the quadratic equation \(a{x}^{2}+bx+c=0\) are from the set \(\left\{1,2,3,4,5,6\right\}\). If the probability of this equation having one real root bigger than the other is \(p\), then \(216p\) equals :
[JEE Main 2024, 5 Apr (Shift 2)]
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Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \(29 / 45\), then \(n\) is equal to :
[JEE Main 2025]
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If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \({i}^{\text{th }}\) roll than the number obtained in the \((i-1{)}^{\text{th }}\) roll, \(i=2,3\), is equal to
[JEE Main 2024, 09 Apr (Shift 2)]
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If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \({i}^{\text{th }}\) roll than the number obtained in the \((i-1{)}^{\text{th }}\) roll, \(i=2,3\), is equal to
[JEE Main 2024, 09 Apr (Shift 2)]
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The probabilities that players \(A\) and \( B\) of a team are selected for the captaincy for a tournament are \(0.6\) and \(0.4\), respectively. If \(A\) is selected as the captain, the probability that the team wins the tournament is \(0.8\) and if \(B\) is selected the captain, the probability that the team wins the tournament is \(0.7\). Then the probability, that the team wins the tournament, is:
[JEE Main 2026, 5 Apr (Shift 2)]
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Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is
[JEE Main 2024, 31 Jan (Shift 1)]
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Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is
[JEE Main 2024, 31 Jan (Shift 1)]
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A bag has 7 good and 3 defected oranges. 2 oranges are chosen randomly. Find the variance of oranges being defected. (28 Jan, Shift I, Memory Based)
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The probability of getting 10 in a single throw of three fair dice is:
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\(\text{ If }{k}_{1},{k}_{2}\in z\text{, then find probability of }{i}^{{k}_{1}}+{i}^{{k}_{2}}\text{ is non-zero }\) (28 Jan, Shift I, Memory Based)
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The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is:
[JEE Main 2025, 4 Apr (Shift 1)]
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Bag \(B_1\) contains 6 white and 4 blue balls, Bag \(B_2\) contains 4 white and 6 blue balls, and Bag \(B_3\) contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag \(B_2\), is :
[JEE Main 2025, 28 Jan (Shift 2)]
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A bag has 7 good and 3 defected oranges. 2 oranges are chosen randomly. Find the variance of oranges being defected. (28 Jan, Shift I, Memory Based)
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There are 3 bags \({B}_{1},{B}_{2},{B}_{3}\) which contain 4 white and 6 black, 6 white and 4 black, 5 white and 5 black balls respectively If 1 ball is randomly selected from a bag and is found to be white . Find the probability that ball came from bag \({B}_{2}\).
[JEE Main 2025]
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Three students \(S_1, S_2\), and \(S_3\) are given a problem to solve. Consider the following events:
\(U\) : At least one of \(S_1, S_2\), and \(S_3\) can solve the problem,
\(V\) : \(S_1\) can solve the problem, given that neither \(S_2\) nor \(S_3\) can solve the problem,
\(W: S_2\) can solve the problem and \(S_3\) cannot solve the problem,
\(T: S_3\) can solve the problem.
For any event \(E\), let \(P(E)\) denote the probability of \(E\). If
\(P(U)=\frac{1}{2},P(V)=\frac{1}{10},\text{and}P(W)=\frac{1}{12}\)
then \(P(T)\) is equal to
[JEE Advanced 2025]
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From a lot containing \(10\) defective and \(90\) non- defective bulbs, \(8\) bulbs are selected one by one with replacement. Then the probability of getting at least \(7\) defective bulbs is
[JEE Main 2026, 24 Jan (Shift 1)]
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Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :
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Two persons \(A\) and \(B\) throws a pair of dice alternatively. For \(A\) to win he should throw sum of 5 before \(B\) throws sum of 8 . If \(A\) throws first, then the probability that \(A\) wins, is (24 Jan, Shift I, Memory Based)
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If \(\ S \) is a set of words formed by all the letters of word "GARDEN", then find the probability that all vowels are not in alphabatical order:
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Let \(A, B\) and \(C\) be three events such that the probability that exactly one of \(A\) and \(B\) occurs is \((1-k)\), the probability that exactly one of \(B\) and \(C\) occurs is \((1-2 k)\), the probability that exactly one of \(C\) and \(A\) occurs is \((1-k)\) and the probability of all \(A , B\) and \(C\) occur simultaneously is \(k^2\), where \(0 [JEE Main 2021, 20 Jul (Shift 2)]
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Let 9 distinct balls be distributed among 4 boxes, \(B_1, B_2\), \(B_3\) and \(B_4\). If the probability that \(B_3\) contains exactly 3 balls is \(k\left(\frac{3}{4}\right)^9\) then \(k\) lies in the set:
[JEE Main 2021, 25 Jul (Shift 1)]
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Let \(S=\left\{w_1, w_2, \ldots \ldots\right\}\) be the sample space associated to a random experiment. Let \(P\left(w_n\right)=\frac{P\left(w_{n-1}\right)}{2}, n \geq 2\).
Let \(A=\{2 k+3 l ; k, l \in N \}\) and \(B=\left\{w_n ; n \in A\right\}\). Then \(P(B)\) is equal to
[JEE Main 2023, 29 Jan (Shift 2)]
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If \(\mathbf{P}(\mathbf{B})=\frac{3}{5}, \mathbf{P}(\mathbf{A} \mid \mathbf{B})=\frac{1}{2}\) and \(\mathbf{P}(\mathbf{A} \cup \mathbf{B})=\frac{4}{5}\), then \(\mathbf{P}(\mathbf{A} \cup \mathbf{B})^{\prime}+\mathbf{P}\left(\mathbf{A}^{\prime} \cup \mathbf{B}\right)=\)
[JEE Main 2023]
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Let a die be rolled \(n\) times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is \(\frac{k}{2^{15}}\), then \(k\) is equal to:
[JEE Main 2023, 10 Apr (Shift 2)]
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The probabilities of three events \( A, B \) and \( C \) are given by \( P(A)=0.6, P(B)=0.4 \) and \( P(C)=0.5 \). If \( P(A \cup B)=0.8, P(A \cap C)=0.3, P(A \cap B \cap C)=0.2 \), \( P(B \cap C)=\beta \) and \( P(A \cup B \cup C)=\alpha \), where \( 0.85 \leq \alpha \leq 0.95 \), then \( \beta \) lies in the interval
[JEE Main 2020, 6 Sep (Shift 2)]
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Bag P contains 6 red and 4 blue balls and bag Q contains 5 red and 6 blue balls. A ball is transferred from bag P to bag Q and then a ball is drawn from bag Q. What is the probability that the ball drawn is blue?
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Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50 . A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :
[JEE Main 2020, 2 Sep (Shift 1)]
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The probability that a card drawn from a pack of 52 cards will be a diamond or king is:
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Let \(N\) denote the number that turns up when a fair die is rolled. If the probability that the system of equations
\(\begin{aligned}& x+y+z=1 \\& 2 x+N y+2 z=2 \\& 3 x+3 y+N z=3\end{aligned}\)
has unique solution is \(\frac{k}{6}\), then the sum of value of \(k\) and all possible values of \(N\) is
[JEE Main 2023, 24 Jan (Shift 1)]
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Each of the persons \(A\) and \(B\) independently tosses three fair coins. The probability that both of them get the same number of heads is:
[JEE Main 2021, 27 Aug (Shift 2)]
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In a binomial distribution \(B(n, P)\), the sum and product of the mean and variance are 5 and 6 respectively, then find \(6(n+p-q)\) is equal to
[JEE Main 2023, 1 Feb (Shift 1)]
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Let \(S=\{1,2,3,4,5,6\}\). Then the probability that a randomly chosen onto function \(g\) from \(S\) to \(S\) satisfies \(g(3)\) \(=2 g(1)\) is:
[JEE Main 2021, 31 Aug (Shift 2)]
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Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter \( M \) appears at the fourth position in any such word is:
[JEE Main 2021, 20 Jul (Shift 1)]
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Let X be a random variable such that the probability function of a distribution is given by \(P(X=0)=\frac{1}{2},P(X=j)=\frac{1}{{3}^{j}}(j=1,2,3,\ldots ,\infty )\). Then the mean of the distribution and \(P(X\) is positive and even) respectively are :
[JEE Main 2021, 25 Jul (Shift 2)]
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If the probability that the random variable \(X\) takes values \(x\) given by \(P\left(X=x\right)=k\left(x+1\right){3}^{-x},x=0,1,2,3,....\) where \(k\) is a constant then \(P\left(X\geq 2\right)\) is equal to
[JEE Main 2023, 8 Apr (Shift 2)]
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Let \(M\) be the maximum value of the product of two positive integers when their sum is 66 . Let the sample space \(S=\left\{x \in Z: x(66-x) \geq \frac{5}{9} M\right\}\) and the event \(A=\{x \in S: x\) is a multiple of 3\(\}\). Then \(P(A)\) is equal to
[JEE Main 2023, 25 Jan (Shift 1)]
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The mean and variance of a random variable \(\ X \) having binomial distribution are 4 and 2 respectively, then \(\ P(X=1) \) is
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Let 9 distinct balls be distributed among 4 boxes, \(B_1, B_2\), \(B_3\) and \(B_4\). If the probability that \(B_3\) contains exactly 3 balls is \(k\left(\frac{3}{4}\right)^9\) then \(k\) lies in the set:
[JEE Main 2021, 25 Feb (Shift 1)]
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A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is:
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The probability that a randomly selected \(2-\)digit number belongs to the set \(\left\{n \in N:\left(2^n-2\right)\right.\) is a multiple of \(3\}\) is equal to:
[JEE Main 2021, 27 Jul (Shift 1)]
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A random variable \(X\) has the following probability distribution:
$$\begin{array}{|c|c|c|c|c|c|}\hline \boldsymbol{X} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\\hline \boldsymbol{P}(\boldsymbol{x}) & k^2 & 2 k & k & 2 k & 5 k^2 \\\hline\end{array}$$
Then \(P(x>2)\) is equal to:
[JEE Main 2020, 9 Jan (Shift 2)]
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Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to:
[JEE Main 2021, 18 Mar (Shift 2)]
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The random variable \(X\) follows binomial distribution \(B(n, p)\) for which the difference of the mean and the variance is \(1\). If \(2 P(X=2)=3 P(X=1)\), then \(n^2 P(X>1)\) is equal to
[JEE Main 2023, 13 Apr (Shift 2)]
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Let \(C_1\) and \(C_2\) be two biased coins such that the probabilities of getting head in a single toss are \(\frac{2}{3}\) and \(\frac{1}{3}\), respectively. Suppose \(\alpha\) is the number of heads that appear when \(\mathrm{C}_1\) is tossed twice, independently, and suppose \(\beta\) is the number of heads that appear when \(\mathrm{C}_2\) is tossed twice, independently, Then the probability that the roots of the quadratic polynomial \(x^2-\alpha x+\beta\) are real and equal, is
[JEE Advanced 2020]
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The probability of selecting integers \(a \in[-5,30]\) such that \(x^2+2(a+4) x-5 a+64>0\), for all \(x \in R\) is:
[JEE Main 2021, 20 Jul (Shift 1)]
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Two dice are thrown independently. Let \(A\) be the event that the number appeared on the \(1^{\text {st }}\) die is less than the number appeared on the \(2^{\text {nd }}\) die, \(B\) be the event that the number appeared on the \(1^{\text {st }}\) die is even and that on the second die is odd, and \(C\) be the event that the number appeared on the \(1^{\text {st }}\) die is odd and that on the \(2^{\text {nd }}\) is even. Then
[JEE Main 2023, 1 Feb (Shift 2)]
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The probability that a randomly selected 2-digit number belongs to the set \(\left\{n \in N:\left(2^n-2\right)\right.\) is a multiple of 3\(\}\) is equal to:
[JEE Main 2021, 27 Jul (Shift 1)]
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Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is \(\frac{1}{3}\), then the probability that the experiment stops with head is.
[JEE Advanced 2023]
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Three dice are rolled. If the probability of getting different numbers on the three dice is \(\frac{p}{q}\), where \(p\) and \(q\) are coprime, then \(q-p\) is equal to
[JEE Main 2023, 6 Apr (Shift 2)]
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Fifteen football players of a club-team are given \(15\) \(T\)-shirts with their names written on the backside. If the players pick up the \(T\)-shirts randomly, then the probability that at least \(3\) players pick the correct \(T\)-shirt is
[JEE Main 2023, 29 Jan (Shift 1)]
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A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is \(\frac{k}{3^{11}}\), then \(k\) is equal to
[JEE Main 2023, 6 Apr (Shift 1)]
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Let \(S=\left\{w_1, w_2, \ldots \ldots\right\}\) be the sample space associated to a random experiment. Let \(P\left(w_n\right)=\frac{P\left(w_{n-1}\right)}{2}, n \geq 2\).Let \(A=\{2 k+3 l ; k, l \in N \}\) and \(B=\left\{w_n ; n \in A\right\}\). Then \(P(B)\) is equal to
[JEE Main 2023, 29 Jan (Shift 2)]
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Girl students constitute \(\ 10 \% \) of I year and \(\ 5 \% \) of II year at Roorkee University. During summer holidays \(\ 70 \% \) of the I year and 30\(\%\) of II year students are given a project. The girls take turns on duty in canteen. The chance that I year girl student is on duty in a randomly selected day is
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An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is
[JEE Main 2021, 24 Feb (Shift 1)]
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Let \(A\) denote the event that a 6 -digit integer formed by 0 , \(1,2,3,4,5,6\) without repetitions, be divisible by 3 . Then probability of event \(A\) is equal to:
[JEE Main 2021, 16 Mar (Shift 2)]
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Let \(N\) denotes the sum of the numbers obtained when two dice are rolled. If the probability that \(2^N [JEE Main 2023, 10 Apr (Shift 1)]
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Let \(S=\left\{M=\left[a_{i j}\right], a_{i j} \in\{0,1,2\}, 1 \leq i, j \leq 2\right\}\) be a sample space and \(A=\{M \in S: M\) is invertible \(\}\) be an event. Then \(P(A)\) is equal to
[JEE Main 2023, 11 Apr (Shift 1)]
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The probability of getting sum more than 7 when a pair of dice are thrown is:
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Two dices are rolled. If both dices have six faces numbered 1,2,3,5,7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is:
[JEE Main 2021, 17 Mar (Shift 1)]
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Let \(N\) be the sum of the numbers appeared when two fair dice are rolled and let the probability that \(N-2, \sqrt{3 N}, N+2\) are in geometric progression be \(\frac{k}{48}\). Then the value of \(k\) is
[JEE Main 2023, 25 Jan (Shift 2)]
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In a bolt factory, machines \(A, B\) and \(C\) manufacture respectively \(20 \%, 30 \%\) and \(50 \%\) of the total bolts. Of their output \(3, 4\) and \(2\) percent are respectively defective bolts. A bolts is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine \(C\) is
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Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the \(T\)-shirts randomly, then the probability that at least 3 players pick the correct \(T\)-shirt is
[JEE Main 2023, 29 Jan (Shift 1)]
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Four dice are thrown simultaneously and the numbers shown on these dice are recorded in \(2 \times 2\) matrices. The probability that such formed matrices have all different entries and are non-singular, is :
[JEE Main 2021, 22 Jul (Shift 2)]
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If an unbiased die, marked with \(-2,-1,0,1,2,3\) on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is :
[JEE Main 2023, 30 Jan (Shift 1)]
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The probability that a missiles hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is
[JEE Advanced 2020]
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A student appeared in an examination consisting of 8 true - false type questions. The student guesses the answers with equal probability. The smallest value of \(n\), so that the probability of guessing at least ' \(n\) ' correct answers is less than \(\frac{1}{2}\), is
[JEE Main 2021, 27 Jul (Shift 2)]
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Let \(X\) be a random variable such that the probability function of a distribution is given by \(P(X=0)=\frac{1}{2}\), \(P(X=j)=\frac{1}{3^j}(j=1,2,3, \ldots, \infty)\). Then, the mean of the distribution and \(P(X)\) is positive and even) respectively are:
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A fair die is tossed until six is obtained on it. Let \(X\) be the number of required tosses, then the conditional probability \(P(X \geq 5 \mid X>2)\) is:
[JEE Main 2021, 26 Aug (Shift 2)]
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Fifteen football players of a club-team are given \(15 T\)-shirts with their names written on the backside. If the players pick up the \(T\)-shirts randomly, then the probability that at least 3 players pick the correct \(T\)-shirt is
[JEE Main 2023, 29 Jan (Shift 1)]
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Words with or without meaning are to be formed using all the letters of the word \(\text{EXAMINATION}\). The probability that the letter \( M \) appears at the fourth position in any such word is:
[JEE Main 2021, 20 Jul (Shift 1)]
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Let \(A\) and \(B\) be two independent events such that \(P(A)\) \(=\frac{1}{3}\) and \(P(B)=\frac{1}{6}\). Then, which of the following is TRUE?
[JEE Main 2020, 8 Jan (Shift 1)]
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Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be \(\frac{1}{2}\) and probability of occurrence of 0 at the odd place be \(\frac{1}{3}\). Then the probability that ' 10 ' is followed by ' 01 ' is equal to:
[JEE Main 2021, 17 Mar (Shift 2)]
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Let \(N\) be the sum of the numbers appeared when two fair dice are rolled and let the probability that \(N-2, \sqrt{3 N}, N+2\) are in geometric progression be \(\frac{k}{48}\). Then the value of \(k\) is
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The probability of selecting integers \(a \in[-5,30]\) such that \(x^2+2(a+4) x-5 a+64>0\), for all \(x \in R\) is:
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The probability that two randomly selected subsets of the set \(\{1,2,3,4,5\}\) have exactly two elements in their intersection is:
[JEE Main 2021, 24 Feb (Shift 2)]
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In a bolt factory, machines \(A, B\) and \(C\) manufacture respectively \(20 \%, 30 \%\) and \(50 \%\) of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolts is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine \(C\) is
[JEE Main 2023, 8 Apr (Shift 1)]
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If a die is thrown twice, then the probability of getting 1 in the first throw only is
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Let \(S = \{ 1,2,3,4,5,6\}\). Then the probability that a randomly chosen onto function \(g\) from \(S\) to \(S\) satisfies \(g(3) = 2g(1)\) is:
[JEE Main 2021, 31 Aug (Shift 2)]
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Let N denote the sum of the numbers obtained when two dice are rolled. If the probability that 2N < N! is \(\frac{m}{n},\) where m and n are coprime, then 4m – 3n equal to:
[JEE Main 2023, 10 Apr (Shift 1)]
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Let \(E^C\) denote the complement of an event \(E\). Let \(E_1\), \(E_2\) and \(E_3\) be any pairwise independent events with \(P\left(E_1\right)>0\) and \(P\left(E_1 \cap E_2 \cap E_3\right)=0\). Then \(P\left( E _2^{ C } \cap E _3^{ C } / E _1\right)\) is equal to:
[JEE Main 2020, 2 Sep (Shift 2)]
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Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is :
[JEE Main 2021, 20 Jul (Shift 1)]
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Bag \(P\) contains 6 red and 4 blue balls and bag \(Q\) contains 5 red and 6 blue balls. A ball is transferred from bag \(P\) to bag \(Q\) and then a ball is drawn from bag \(Q\). What is the probability that the ball drawn is blue?
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A bag contains \(6\) white and \(4\) black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is:
[JEE Main 2023, 15 Apr (Shift 1)]
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When a certain biased die is rolled, a particular face occurs with probability \(\frac{1}{6}-x\) and its opposite face occurs with
probability \(\frac{1}{6}+x\). All other faces occur with probability \(\frac{1}{6}\). Note that opposite faces sum to 7 in any die. If \(0 [JEE Main 2021, 27 Aug (Shift 1)]
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Let \(A\) denote the event that a 6 -digit integer formed by \(0,1,2,3,4,5,6\) without repetitions, be divisible by 3 . Then probability of event \(A\) is equal to:
[JEE Main 2021, 16 Mar (Shift 2)]
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If the probability that random variable X takes values x is given by P (X = x) = k (x + 1)3–x, x = 0, 1, 2, 3, …., where k is a constant, then P (X \(\geq\) 2) is equal to
[JEE Main 2023, 08 Apr (Shift 2)]
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In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine C is.
[JEE Main 2023, 08 Apr (Shift 1)]
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The probability of selecting integers \(a \in[-5,30]\) such that \(x^2+2(a+4) x-5 a+64>0\), for all \(x \in R\) is :
[JEE Main 2021, 20 Jul (Shift 1)]
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The probabilities of three events \( A, B \) and \( C \) are given by \( P(A)=0.6, P(B)=0.4 \) and \( P(C)=0.5 \) If \( P(A \cup B)=0.8, P(A \cap C)=0.3, P(A \cap B \cap C)=0.2 \), \( P(B \cap C)=\beta \) and \( P(A \cup B \cup C)=\alpha \), where \( 0.85 \leq \alpha \leq 0.95 \), then \( \beta \) lies in the interval
[JEE Main 2020, 6 Sep (Shift 2)]
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