JEE Main Previous Year Probability Questions and Solutions | Addition Theorem, Conditional Probability and Bayes’ Theorem

Q 1 :
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 urns 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 __________ . [2024]

$\frac{5}{16}$
$\frac{4}{17}$
$\frac{5}{18}$
$\frac{7}{18}$

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(3)

Let A, B, C and E be the events defined as follows

A: First urn is chosen

B: Second urn is chosen

C: Third urn is chosen

E: Ball drawn is black

$∴$ $P (A) = P (B) = P (C) = \frac{1}{3}$

$∴$ $P (E | A) = \frac{5}{12}, P (E | B) = \frac{7}{12}, P (E | C) = \frac{6}{12}$

By Bayes' Theorem,

$P (A | E) = \frac{P (A) \cdot P (E | A)}{P (A) \cdot P (E | A) + P (B) \cdot P (E | B) + P (C) \cdot P (E | C)}$

$= \frac{\frac{1}{3} \times \frac{5}{12}}{\frac{1}{3} \times \frac{5}{12} + \frac{1}{3} \times \frac{7}{12} + \frac{1}{3} \times \frac{6}{12}} = \frac{5}{5 + 7 + 6} = \frac{5}{18}$

Q 2 :
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 [2024]

64
56
66
54

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(4)

Let $E_{1}$ be the event that motorcycle manufactured in plant A.

$E_{2}$ be the event that motorcycle manufactured in plant B, and E be the event that it found to be of standard quality.

$∴$ $P (E_{1}) = \frac{60}{100}, P (E_{2}) = \frac{40}{100}$

$P (\frac{E}{E_{1}}) = \frac{80}{100}$ and $P (\frac{E}{E_{2}}) = \frac{90}{100}$

Now, $p = P (\frac{E_{2}}{E}) = \frac{P (E | E_{2}) \cdot P (E_{2})}{P (E | E_{2}) \cdot P (E_{2}) + P (E | E_{1}) \cdot (E_{1})}$

$= \frac{\frac{90}{100} \cdot \frac{40}{100}}{\frac{60}{100} \cdot \frac{80}{100} + \frac{90}{100} \cdot \frac{40}{100}} = \frac{36}{48 + 36} = \frac{3}{7}$

$∴$ $126 p = 126 \times \frac{3}{7} = 54$

Q 3 :
If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is ______. [2024]

$\frac{6}{25}$
$\frac{4}{25}$
$\frac{18}{25}$
$\frac{12}{25}$

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(4)

We need to choose 2 addresses out of 5 which can be done in ${}^{5}C_{2}$ ways.

Now, 3 letters can be posted to exactly 2 addresses in $3 \times 2 = 6$ ways.

Required Probability = $\frac{{}^{5}{C_{2}} \times 6}{5^{3}} = \frac{60}{125} = \frac{12}{25}$

Q 4 :
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 $g c d (m, n) = 1,$ then $n - m$ equals [2024]

9
10
11
8

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(2)

Let the two positive integers be $x$ and $y$ .

Now, $x + y = 24,$ $x, y \in N$

Now, $\frac{x + y}{2} \geq \sqrt{x y}$ $[∵ A . M . \geq G . M .]$

$\Rightarrow x y \leq 144$

Since, $\frac{3}{4} \times 144 = 108$

So, $x y \geq 108$

$∴$ Favourable cases = ${(6, 18), (18, 6), (7, 17), (17, 7), (8, 16), (16, 8), (9, 15), (15, 9), (10, 14), (14, 10), (11, 13), (13, 11), (12, 12)}$

Total choices for $x + y = 24$ are 23

$∴$ Required probability $= \frac{13}{23} = \frac{m}{n}$

$∴$ $n - m = 23 - 13 = 10$

Q 5 :
There are three bags X, Y, and Z. Bag X contains 5 one-rupee coins and 4 five-rupee coins; Bag Y contains 4 one-rupee coins and 5 five-rupee coins, and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability that it came from bag Y is [2024]

$\frac{1}{2}$
$\frac{5}{12}$
$\frac{1}{4}$
$\frac{1}{3}$

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(4)

Let $E_{1}, E_{2}, E_{3}$ be the event of selecting bags X, Y and Z respectively and A be the event that coin drawn is one-rupee coin.

$∴$ $P (E_{1}) = P (E_{2}) = P (E_{3}) = \frac{1}{3}$

$P (A | E_{1}) = \frac{5}{9}, P (A | E_{2}) = \frac{4}{9}, P (A | E_{3}) = \frac{3}{9}$

By Bayes' theorem,

Required Probability = $P (E_{2} | A) = \frac{\frac{1}{3} \times \frac{4}{9}}{\frac{1}{3} \times \frac{5}{9} + \frac{1}{3} \times \frac{4}{9} + \frac{1}{3} \times \frac{3}{9}}$

$= \frac{4}{5 + 4 + 3} = \frac{4}{12} = \frac{1}{3}$

Q 6 :
If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{t h}$ roll than the number obtained in the ${(i - 1)}^{t h}$ roll, $i = 2, 3,$ is equal to [2024]

$\frac{5}{54}$
$\frac{2}{54}$
$\frac{1}{54}$
$\frac{3}{54}$

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(1)

Let X be the event of getting a greater number than the previous one in a throw of a die.

Favourable outcomes to $X = {}^{6}{C_{3}}$

[ $∵$ ${}^{6}{C_{3}}$ are ways of getting 3 outcomes one less than other as we have total 6 possible outcomes]

Total outcomes = $6^{3}$

$P (X) = \frac{{}^{6}{C_{3}}}{6^{3}} = \frac{20}{216} = \frac{5}{54}$

Q 7 :
A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is [2024]

$\frac{2}{5}$
$\frac{1}{7}$
$\frac{1}{5}$
$\frac{2}{7}$

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(4)

The balls can be distributed as (2W, 6B), (3W, 5B), (4W, 4B), (5W, 3B), (6W, 2B)

$∴$ $Required probability$ $= \frac{\frac{{}^{4}C_{2} \times {}^{4}{C_{2}}}{\frac{{}^{8}{C_{4}}}{2 ({}^{2}{C_{2} \times {}^{6}{C_{2}}}) + (2) ({}^{3}{C_{2} \times {}^{5}{C_{2}}}) + {}^{4}{C_{2}} \times {}^{4}{C_{2}}}}}{{}^{8}{C_{4}}}$

$= \frac{6 \times 6}{2 (15) + 2 (30) + 36} = \frac{36}{30 + 60 + 36} = \frac{36}{126} = \frac{2}{7}$

Q 8 :
Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7},$ while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5} .$ Then the probability that Ajay will appear in the exam and Vijay will not appear is [2024]

$\frac{9}{35}$
$\frac{3}{35}$
$\frac{24}{35}$
$\frac{18}{35}$

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(4)

We have, $P (Not Ajay) = P = \frac{2}{7} \Rightarrow P (Ajay) = P = \frac{5}{7}$

$P (Ajay \cap Vijay) = q = \frac{1}{5}$

$∴$ $P (Ajay \cap not Vijay) = \bar{p} - q = \frac{5}{7} - \frac{1}{5} = \frac{18}{35}$

Q 9 :
An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls and the second draw gives all black balls, is [2024]

$\frac{5}{715}$
$\frac{5}{256}$
$\frac{3}{715}$
$\frac{3}{256}$

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(3)

Let A : event when $1^{s t}$ draw gives all 4 white balls.

and B : event when $2^{n d}$ draw gives all 4 black balls.

$P (A) = \frac{{}^{6}{C_{4}}}{{}^{15}{C_{4}}} = \frac{1}{91}$ and $P (B | A) = \frac{{}^{9}{C_{4}}}{{}^{11}{C_{4}}} = \frac{21}{55}$

$∴$ $Required probability = P (A) \times P (\frac{B}{A}) = \frac{1}{91} \times \frac{21}{55} = \frac{3}{715}$

Q 10 :
A fair die is thrown until 2 appears. Then the probability that 2 appears in an even number of throws, is [2024]

$\frac{1}{6}$
$\frac{6}{11}$
$\frac{5}{6}$
$\frac{5}{11}$

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(4)

Probability that 2 appears $= \frac{1}{6}$

Probability that 2 does not appear $= \frac{5}{6}$

Required probability $= \frac{5}{6} \times \frac{1}{6} + \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} \times \frac{1}{6} + \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} \times \frac{1}{6} + \dots$

$= \frac{5}{6} \times \frac{1}{6} + {(\frac{5}{6})}^{3} \times \frac{1}{6} + {(\frac{5}{6})}^{5} \times \frac{1}{6} + \dots$ $= \frac{\frac{5}{6} \times \frac{1}{6}}{1 - \frac{5}{6} \times \frac{5}{6}}$ $= \frac{5}{11}$

Topic Question Set

Q 1 :
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 urns 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 __________ . [2024]

Q 3 :
If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is ______. [2024]

Q 4 :
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 $g c d (m, n) = 1,$ then $n - m$ equals [2024]

Q 6 :
If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{t h}$ roll than the number obtained in the ${(i - 1)}^{t h}$ roll, $i = 2, 3,$ is equal to [2024]

Q 7 :
A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is [2024]

Q 8 :
Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7},$ while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5} .$ Then the probability that Ajay will appear in the exam and Vijay will not appear is [2024]

Q 9 :
An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls and the second draw gives all black balls, is [2024]

Q 10 :
A fair die is thrown until 2 appears. Then the probability that 2 appears in an even number of throws, is [2024]

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Topic Question Set

Q 1 : 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 urns 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 __________ . [2024]

Q 3 : If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is ______. [2024]

Q 4 : Let the sum of two positive integers be 24. If the probability that their product is not less than 34 times their greatest possible product is mn, where gcd(m,n)=1, then n-m equals [2024]

Q 6 : If an unbiased dice is rolled thrice, then the probability of getting a greater number in the ith roll than the number obtained in the (i-1)th roll, i=2,3, is equal to [2024]

Q 7 : A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is [2024]

Q 8 : Let Ajay will not appear in JEE exam with probability p=27, while both Ajay and Vijay will appear in the exam with probability q=15. Then the probability that Ajay will appear in the exam and Vijay will not appear is [2024]

Q 9 : An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls and the second draw gives all black balls, is [2024]

Q 10 : A fair die is thrown until 2 appears. Then the probability that 2 appears in an even number of throws, is [2024]

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Q 1 :
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 urns 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 __________ . [2024]

Q 3 :
If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is ______. [2024]

Q 4 :
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 $g c d (m, n) = 1,$ then $n - m$ equals [2024]

Q 6 :
If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{t h}$ roll than the number obtained in the ${(i - 1)}^{t h}$ roll, $i = 2, 3,$ is equal to [2024]

Q 7 :
A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is [2024]

Q 8 :
Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7},$ while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5} .$ Then the probability that Ajay will appear in the exam and Vijay will not appear is [2024]

Q 9 :
An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls and the second draw gives all black balls, is [2024]

Q 10 :
A fair die is thrown until 2 appears. Then the probability that 2 appears in an even number of throws, is [2024]