Bayes' Theorem

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JEE Advanced 2016

LEVELJEE Main

A computer producing factory has only two plants $T_{1}$ and $T_{2}$ . Plant $T_{1}$ produces 20% and plant $T_{2}$ produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that $P (computer turns out to be defective given that it is produced in plant T_{1}) = 10 P (computer turns out to be defective given that it is produced in plant T_{2})$ , where $P (E)$ denotes the probability of an event $E$ . A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_{2}$ is

A signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is $\frac{3}{4}$ . If the signal received at station B is green, then the probability that the original signal was green is

Let $H_{1}, H_{2}, \dots, H_{n}$ be mutually exclusive and exhaustive events with $P (H_{i}) > 0, i = 1, 2, \dots, n$ . Let $E$ be any other event with $0 < P (E) < 1$ . STATEMENT-1: $P (H_{i} ∣ E) > P (E ∣ H_{i}) \cdot P (H_{i})$ for $i = 1, 2, \dots, n$ because STATEMENT-2: $\sum_{i = 1}^{n} P (H_{i}) = 1$ .

(A)

Statement-1 is True, statement-2 is True; Statement-2 is a correct explanation for Statement-1.

(B)

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

(C)

Statement-1 is True, Statement-2 is False.

(D)

Statement-1 is False, Statement-2 is True.

JEE Advanced 2005

LEVELJEE Main

A person goes to office either by car, scooter, bus or train, the probability of which being $1/7, 3/7, 2/7$ and $1/7$ respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is $2/9, 1/9, 4/9$ and $1/9$ respectively. Given that he reached office in time, then what is the probability that he travelled by a car.

JEE Advanced 2002

LEVELJEE Main

A box contains $N$ coins, $m$ of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is $1/2$ , while it is $2/3$ when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows head and the second time it shows tail. What is the probability that the coin drawn is fair?

JEE Advanced 1991

LEVELJEE Main

In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he make a guess is $1/3$ and the probability that he copies the answer is $1/6$ . The probability that his answer is correct given that he copied it, is $1/8$ . Find the probability that he knew the answer to the question given that he correctly answered it.

JEE Advanced 2015

LEVELJEE Advanced

Comprehension Passage

Let

n_{1}

and

n_{2}

be the number of red and black balls, respectively, in box I. Let

n_{3}

and

n_{4}

be the number of red and black balls, respectively, in box II.

Question 1:

One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is 1/3, then the correct option(s) with the possible values of $n_{1}, n_{2}, n_{3}$ and $n_{4}$ is(are)

* Multiple Correct Options

(A)

n_{1} = 3, n_{2} = 3, n_{3} = 5, n_{4} = 15

(B)

n_{1} = 3, n_{2} = 6, n_{3} = 10, n_{4} = 50

(C)

n_{1} = 8, n_{2} = 6, n_{3} = 5, n_{4} = 20

(D)

n_{1} = 6, n_{2} = 12, n_{3} = 5, n_{4} = 20

Question 2:

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is 1/3, then the correct option(s) with the possible values of $n_{1}$ and $n_{2}$ is(are)

* Multiple Correct Options

(A)

n_{1} = 4

and

n_{2} = 6

(B)

n_{1} = 2

and

n_{2} = 3

(C)

n_{1} = 10

and

n_{2} = 20

(D)

n_{1} = 3

and

n_{2} = 6

JEE Advanced 2013

LEVELJEE Main

Comprehension Passage

A box

B_{1}

contains 1 white ball, 3 red balls and 2 black balls. Another box

B_{2}

contains 2 white balls, 3 red balls and 4 black balls. A third box

B_{3}

contains 3 white balls, 4 red balls and 5 black balls.

Question 1:

If 1 ball is drawn from each of the boxes $B_{1}, B_{2}$ and $B_{3}$ , the probability that all 3 drawn balls are of the same colour is

If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box $B_{2}$ is

Comprehension Passage

Let

U_{1}

and

U_{2}

be two urns such that

U_{1}

contains 3 white and 2 red balls, and

U_{2}

contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from

U_{1}

and put into

U_{2}

. However, if tail appears then 2 balls are drawn at random from

U_{1}

and put into

U_{2}

. Now 1 ball is drawn at random from

U_{2}

Question 1:

The probability of the drawn ball from $U_{2}$ being white is

Given that the drawn ball from $U_{2}$ is white, the probability that head appeared on the coin is

Comprehension Passage

There are

n

urns, each of these contain

n + 1

balls. The

i^{t h}

urn contains

i

white balls and

(n + 1 - i)

red balls. Let

u_{i}

be the event of selecting

i^{t h}

urn,

i = 1, 2, 3, \dots, n

and

w

the event of getting a white ball.

Question 1:

If $P (u_{i}) \propto i$ , where $i = 1, 2, 3, \dots, n$ , then $lim_{n \to \infty} P (w) =$

If $P (u_{i}) = c$ , (a constant) then $P (u_{n} / w) =$

(A)

\frac{2}{n + 1}

(B)

\frac{1}{n + 1}

(C)

\frac{n}{n + 1}

(D)

\frac{1}{2}

Question 3:

Let $P (u_{i}) = \frac{1}{n}$ , if $n$ is even and $E$ denotes the event of choosing even numbered urn, then the value of $P (w / E)$ is

(A)

\frac{n + 2}{2 n + 1}

(B)

\frac{n + 2}{2 ( n + 1 )}

(C)

\frac{n}{n + 1}

(D)

\frac{1}{n + 1}

JEE Main 2018 (15 April Shift 1)

LEVELJEE Main

A box 'A' contains 2 white, 3 red and 2 black balls. Another box 'B' contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box 'B' is :

JEE Main 2020 - 2 Sep (Morning)

LEVELJEE Main

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 1 is :

JEE Main 2021 (31 Aug Shift 1)

LEVELJEE Main

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8. The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $p$ , then $98 p$ is equal to .

JEE Main 2021 (25 February Shift 2)

LEVELJEE Main

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35%$ , $20%$ and $10%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:

(A)

\frac{7}{45}

(B)

\frac{8}{45}

(C)

\frac{14}{45}

(D)

\frac{28}{45}

JEE Main 2021 (March)

LEVELJEE Main

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 :

(A)

3/4

(B)

52/867

(C)

39/50

(D)

22/425

JEE Main 2022 (24 June Shift 1)

LEVELJEE Main

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and $n$ white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is $\frac{6}{11}$ , then $n$ is equal to ____.

JEE Main 2022 (29 July Shift 2)

LEVELJEE Main

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is draw from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is:

(A)

\frac{4}{9}

(B)

\frac{5}{18}

(C)

\frac{1}{6}

(D)

\frac{3}{10}

JEE Main 2023 (24 January Shift 2)

LEVELJEE Advanced

The 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

JEE Main 2023 (08 April Shift 1)

LEVELJEE Main

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

(A)

\frac{5}{14}

(B)

\frac{9}{28}

(C)

\frac{3}{7}

(D)

\frac{2}{7}

JEE Main 2023 (25 January Shift 2)

LEVELJEE Main

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

JEE Main 2023 (31 January Shift 1)

LEVELJEE Main

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

(A)

\frac{5}{7}

(B)

\frac{2}{7}

(C)

\frac{3}{7}

(D)

\frac{5}{6}

JEE Main 2024 (04 Apr Shift 1)

LEVELJEE Main

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 :

(A)

\frac{5}{18}

(B)

\frac{5}{16}

(C)

\frac{4}{17}

(D)

\frac{7}{18}

JEE Main 2024 (06 Apr Shift 1)

LEVELJEE Main

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 (08 Apr Shift 2)

LEVELJEE Main

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 :

(A)

\frac{1}{4}

(B)

\frac{1}{2}

(C)

\frac{5}{12}

(D)

\frac{1}{3}

JEE Main 2024 (01 Feb Shift 1)

LEVELJEE Main

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:

(A)

\frac{2}{5}

(B)

\frac{2}{7}

(C)

\frac{1}{7}

(D)

\frac{1}{5}

Comprehension Passage

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 1 ball is drawn from each of the boxes B1​,B2​ and B3​, the probability that all 3 drawn balls are of the same colour is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability of the drawn ball from U2​ being white is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given that the drawn ball from U2​ is white, the probability that head appeared on the coin is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P(ui​)∝i, where i=1,2,3,…,n, then limn→∞​P(w)=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P(ui​)=c, (a constant) then P(un​/w)=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P(ui​)=n1​, if n is even and E denotes the event of choosing even numbered urn, then the value of P(w/E) is

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 1 is :

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 :

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is draw from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is:

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

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 :

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:

If 1 ball is drawn from each of the boxes $B_{1}, B_{2}$ and $B_{3}$ , the probability that all 3 drawn balls are of the same colour is

The probability of the drawn ball from $U_{2}$ being white is

Given that the drawn ball from $U_{2}$ is white, the probability that head appeared on the coin is

If $P (u_{i}) \propto i$ , where $i = 1, 2, 3, \dots, n$ , then $lim_{n \to \infty} P (w) =$

If $P (u_{i}) = c$ , (a constant) then $P (u_{n} / w) =$

Let $P (u_{i}) = \frac{1}{n}$ , if $n$ is even and $E$ denotes the event of choosing even numbered urn, then the value of $P (w / E)$ is