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

Q 41 :
Let $M$ be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $S = {x \in Z : x (66 - x) \geq \frac{5}{9} M}$ and the event $A = {x \in S : x is a multiple of 3} .$ Then $P (A)$ is equal to [2023]

$\frac{15}{44}$
$\frac{1}{3}$
$\frac{7}{22}$
$\frac{1}{5}$

1

2

3

4

(2)

$M = 33^{2} (Using A.M. and G.M. inequality)$

$x (66 - x) \geq \frac{5}{9} \times 33^{2}$

$\Rightarrow x^{2} - 66 x + \frac{5}{9} \times 33^{2} \leq 0 \Rightarrow x^{2} - 66 x + 605 \leq 0$ ...(i)

$\Rightarrow (x - 55) (x - 11) \leq 0;$ $11 \leq x \leq 55$

$∴$ $S = {11, 12, 13, \dots, 55} \Rightarrow n (S) = 45$

$Elements of S which are multiples of 3 are$

$12 + (n - 1) 3 = 54 \Rightarrow 3 (n - 1) = 42 \Rightarrow n = 15$

$n (A) = 15 \Rightarrow P (A) = \frac{15}{45} = \frac{1}{3}$

Q 42 :
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 [2023]

16
2
8
4

1

2

3

4

(4)

$Here, n (S) = 36$

$N denotes the sum of numbers when two fair dice are rolled.$

$N - 2, \sqrt{3 N}, N + 2 are in G.P.$

$3 N = (N - 2) (N + 2)$

$\Rightarrow 3 N = N^{2} - 4 \Rightarrow N^{2} - 3 N - 4 = 0$

$\Rightarrow (N - 4) (N + 1) = 0$

$\Rightarrow N = 4$ $(since N = - 1 rejected)$

$Favorable cases: (1, 3), (3, 1), (2, 2)$

$Required probability = \frac{3}{36} = \frac{k}{48} (given) \Rightarrow k = 4$

Q 43 :
Let $S = {w_{1}, w_{2}, \dots}$ be the sample space associated to a random experiment. Let $P (w_{n}) = \frac{P (w_{n - 1})}{2}, n \geq 2 .$ Let $A = {2 k + 3 l : k, l \in ℕ}$ and $B = {w_{n} : n \in A}$ . Then $P (B)$ is equal to [2023]

$\frac{1}{16}$
$\frac{3}{64}$
$\frac{3}{32}$
$\frac{1}{32}$

1

2

3

4

(2)

Given $S = {w_{1}, w_{2}, \dots}$ be the sample space associated to a random experiment. If

$P (W_{n}) = \frac{P (W_{n - 1})}{2}, n \geq 2$

and let $A = {2 k + 3 l : k, l \in N}$ and $B = {w_{n} : n \in A}$ .

First of all, let $P (w_{1}) = λ$ . Then $P (w_{2}) = \frac{λ}{2}$

$P (w_{n}) = \frac{λ}{2^{n - 1}}$

As
$\sum_{k = 1}^{\infty} P (w_{k}) = 1 \Rightarrow \frac{λ}{1 - \frac{1}{2}} = 1 \Rightarrow λ = \frac{1}{2}$

So $P (W_{n}) = \frac{1}{2^{n}}$

$A = {2 k + 3 l : k, l \in N} = {5, 7, 8, 9, 10, \dots}$

$B = {w_{n} : n \in A}$

$B = {w_{5}, w_{7}, w_{8}, w_{9}, w_{10}, w_{11}, \dots}$

$A = N - {1, 2, 3, 4, 6}$

$P (B) = 1 - [P (w_{1}) + P (w_{2}) + P (w_{3}) + P (w_{4}) + P (w_{6})]$

$= 1 - [\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{64}] = 1 - \frac{32 + 16 + 8 + 4 + 1}{64} = \frac{3}{64}$

Q 44 :
A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is [2023]

$\frac{2}{7}$
$\frac{3}{7}$
$\frac{5}{7}$
$\frac{5}{6}$

1

2

3

4

(3)

Total possibilities :

Case I : 2B + 4 others

Case II : 3B + 3 others

Case III : 4B + 2 others

Case IV : 5B + 1 other

Case V : 6B + 0 other

$Required probability = \frac{\frac{{}^{5}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{6}{C_{2}}}{{}^{6}{C_{2}}}}{\frac{{}^{2}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{3}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{4}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{5}{C_{2}}}{{}^{6}{C_{2}}} + \frac{{}^{6}{C_{2}}}{{}^{6}{C_{2}}}}$

$= \frac{10 + 15}{1 + 3 + 6 + 10 + 15} = \frac{25}{35} = \frac{5}{7}$

Q 45 :
Let the probability of getting head for a biased coin be $\frac{1}{4}$ . It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the probability that the equation $64 x^{2} + 5 N x + 1 = 0$ has no real root is $\frac{p}{q}$ , where $p$ and $q$ are co-prime, then $q - p$ is equal to ______. [2023]

0%

(27)

Let $H$ be the event of getting head and $T$ be the event of getting tail. $P (H) = \frac{1}{4} and P (T) = \frac{3}{4}$

Now, $64 x^{2} + 5 N x + 1 = 0$

$D = 25 N^{2} - 4 (64) (1) = 25 N^{2} - 256$

For no real roots, $D < 0$

$\Rightarrow 25 N^{2} - 256 < 0$

$\Rightarrow 25 N^{2} < 256 \Rightarrow N^{2} < \frac{256}{25} \Rightarrow N < \frac{16}{5}$

$∴$ $N = 1, 2, 3$

Required probability $= H + T H + T T H$

$= \frac{1}{4} + \frac{3}{4} \times \frac{1}{4} + \frac{3}{4} \times \frac{3}{4} \times \frac{1}{4}$

$= \frac{1}{4} + \frac{3}{16} + \frac{9}{64} = \frac{37}{64} = \frac{p}{q} (Given)$

$∴$ $q - p = 64 - 37 = 27$

Q 46 :
Three urns A, B and C contain 4 red, 6 black; 5 red, 5 black; and $λ$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0.4, then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola $y^{2} = λ x$ with one vertex at the vertex of the parabola, is __________. [2023]

0%

(432)

$P (Red ball from urn C)$

$=$ $\frac{\frac{1}{3} \cdot \frac{λ}{λ + 4}}{\frac{1}{3} \cdot \frac{4}{10} + \frac{1}{3} \cdot \frac{5}{10} + \frac{1}{3} \cdot \frac{λ}{λ + 4}} = 0.4 \Rightarrow \frac{\frac{λ}{λ + 4}}{\frac{4}{10} + \frac{5}{10} + \frac{λ}{λ + 4}} = \frac{4}{10}$

$∴$ $λ = 6$

So, parabola $y^{2} = 6 x$

Let side length of the triangle be $l$ .

$\tan 30 ° = \frac{3 t}{\frac{3}{2} t^{2}}$

$\Rightarrow \frac{1}{\sqrt{3}} = \frac{2}{t}$

$∴$ $t = 2 \sqrt{3}$

So, $(\frac{3}{2} t^{2}, 3 t) = (18, 6 \sqrt{3})$

Now, $l^{2} = 18^{2} + {(6 \sqrt{3})}^{2} = 324 + 108 = 432$

Q 47 :
25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$ . Then the value of $k$ is _______. [2023]

0%

(9)

Let number of smokers be $E_{1}$ . So, $P (E_{1}) = \frac{1}{4}$ .

The number of non-smokers be $E_{2}$ . So, $P (E_{2}) = \frac{3}{4}$ .

Let E denote persons diagnosed with lung cancer.

Let $P (E / E_{2}) = x, P (E / E_{1}) = 27 x$

Now, $P (E_{1} / E) = \frac{P (E_{1}) P (E / E_{1})}{P (E)}$

$= \frac{\frac{1}{4} \times 27 \times x}{\frac{1}{4} \times 27 \times x + \frac{3}{4} \times x} = \frac{27}{30} = \frac{9}{10}$

So, $k = 9$

Q 48 :
A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p .$ Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is $q$ . If $p : q = m : n$ ,where $m$ and $n$ are coprime, then $m + n$ is equal to _______. [2023]

0%

(14)

Bag contains six balls of different colours.

Probability of drawing a ball of one colour $= \frac{1}{6}$

$∴$ $Probability of drawing two balls of same colour with replacement = p = {(\frac{1}{6})}^{2}$

Probability that exactly three balls are of same colour when four balls are drawn in succession with replacement.

$= q = 4 {(\frac{1}{6})}^{3} \times \frac{5}{6} = \frac{20}{6^{4}}$

$p : q = \frac{1}{6^{2}} : \frac{20}{6^{4}} = \frac{36}{20} = \frac{9}{5}$

$p : q = m : n = 9 : 5$ $∴$ $m = 9, n = 5$

$Now, m + n = 9 + 5 = 14$

Q 49 :
Let A be the event that the absolute difference between two randomly chosen real numbers in the sample space [0, 60] is less than or equal to $a$ .If $P (A) = \frac{11}{36},$ then $a$ is equal to ______ . [2023]

0%

(10)

$| x - y | \leq a$

$\Rightarrow - a \leq x - y \leq a \Rightarrow x - y \leq a and x - y \geq - a$

$P (A) = \frac{ar (O A C D E G)}{ar (O B D F)} = \frac{ar (O B D F) - ar (∆ A B C) - ar (∆ E F G)}{ar (O B D F)}$

$\Rightarrow \frac{11}{36} = \frac{60^{2} - \frac{1}{2} {(60 - a)}^{2} - \frac{1}{2} {(60 - a)}^{2}}{3600}$

$\Rightarrow 1100 = 3600 - {(60 - a)}^{2}$

$\Rightarrow {(60 - a)}^{2} = 2500 \Rightarrow 60 - a = 50 \Rightarrow a = 10$

Topic Question Set

Q 41 :
Let $M$ be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $S = {x \in Z : x (66 - x) \geq \frac{5}{9} M}$ and the event $A = {x \in S : x is a multiple of 3} .$ Then $P (A)$ is equal to [2023]

Q 42 :
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 [2023]

Q 43 :
Let $S = {w_{1}, w_{2}, \dots}$ be the sample space associated to a random experiment. Let $P (w_{n}) = \frac{P (w_{n - 1})}{2}, n \geq 2 .$ Let $A = {2 k + 3 l : k, l \in ℕ}$ and $B = {w_{n} : n \in A}$ . Then $P (B)$ is equal to [2023]

Q 44 :
A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is [2023]

0%

0%

Q 47 :
25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$ . Then the value of $k$ is _______. [2023]

0%

0%

Q 49 :
Let A be the event that the absolute difference between two randomly chosen real numbers in the sample space [0, 60] is less than or equal to $a$ .If $P (A) = \frac{11}{36},$ then $a$ is equal to ______ . [2023]

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Topic Question Set

Q 41 : Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space S={x∈Z:x(66-x)≥59M} and the event A={x∈S:x is a multiple of 3}. Then P(A) is equal to [2023]

Q 42 : Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that N-2, 3N, N+2 are in geometric progression be k48. Then the value of k is [2023]

Q 43 : Let S={w1,w2,…} be the sample space associated to a random experiment. Let P(wn)=P(wn-1)2,n≥2. Let A={2k+3l:k,l∈ℕ} and B={wn:n∈A}. Then P(B) is equal to [2023]

Q 44 : A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is [2023]

Q 45 : Let the probability of getting head for a biased coin be 14. It is tossed repeatedly until a head appears. Let N be the number of tosses required. If the probability that the equation 64x2+5Nx+1=0 has no real root is pq, where p and q are co-prime, then q-p is equal to ______. [2023]

0%

0%

Q 47 : 25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is k10. Then the value of k is _______. [2023]

0%

0%

Q 49 : Let A be the event that the absolute difference between two randomly chosen real numbers in the sample space [0, 60] is less than or equal to a.If P(A)=1136, then a is equal to ______ . [2023]

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Q 41 :
Let $M$ be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $S = {x \in Z : x (66 - x) \geq \frac{5}{9} M}$ and the event $A = {x \in S : x is a multiple of 3} .$ Then $P (A)$ is equal to [2023]

Q 42 :
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 [2023]

Q 43 :
Let $S = {w_{1}, w_{2}, \dots}$ be the sample space associated to a random experiment. Let $P (w_{n}) = \frac{P (w_{n - 1})}{2}, n \geq 2 .$ Let $A = {2 k + 3 l : k, l \in ℕ}$ and $B = {w_{n} : n \in A}$ . Then $P (B)$ is equal to [2023]

Q 44 :
A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is [2023]

Q 47 :
25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$ . Then the value of $k$ is _______. [2023]

Q 49 :
Let A be the event that the absolute difference between two randomly chosen real numbers in the sample space [0, 60] is less than or equal to $a$ .If $P (A) = \frac{11}{36},$ then $a$ is equal to ______ . [2023]