Conditional Probability

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JEE Main 2012

LEVELJEE Main

Three numbers are chosen at random without replacement from ${1, 2, 3, ..., 8}$ . The probability that their minimum is 3, given that their maximum is 6, is

If $C$ and $D$ are two events such that $C \subset D$ and $P (D) \neq = 0$ , then the correct statement among the following is

(A)

P (C ∣ D) \geq P (C)

(B)

P (C ∣ D) < P (C)

(C)

P (C ∣ D) = \frac{P ( D )}{P ( C )}

(D)

P (C ∣ D) = P (C)

JEE Main 2009

LEVELJEE Main

One ticket is selected at random from 50 tickets numbered 00, 01, 02, ..., 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals

It is given that the events $A$ and $B$ are such that $P (A) = 1/4, P (A ∣ B) = 1/2$ and $P (B ∣ A) = 2/3$ . Then $P (B)$ is

A ship is fitted with three engines $E_{1}, E_{2}$ and $E_{3}$ . The engines function independently of each other with respective probabilities $1/2, 1/4$ and $1/4$ . For the ship to be operational at least two of its engines must function. Let $X$ denote the event that the ship is operational and let $X_{1}, X_{2}$ and $X_{3}$ denote respectively the events that the engines $E_{1}, E_{2}$ and $E_{3}$ are functioning. Which of the following is(are) true?

* Multiple Correct Options

(A)

P [X_{1}^{c} ∣ X] = 3/16

(B)

P [Exactly two engines of the ship are functioning ∣ X] = 7/8

(C)

P [X ∣ X_{2}] = 5/16

(D)

P [X ∣ X_{1}] = 7/16

JEE Advanced 2012

LEVELJEE Main

Let $X$ and $Y$ be two events such that $P (X ∣ Y) = 1/2, P (Y ∣ X) = 1/3$ and $P (X \cap Y) = 1/6$ . Which of the following is (are) correct?

* Multiple Correct Options

(A)

P (X \cup Y) = 2/3

(B)

X

and

Y

are independent

(C)

X

and

Y

are not independent

(D)

P (X^{c} \cap Y) = 1/3

JEE Advanced 2007

LEVELJEE Main

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

Let $E^{c}$ denote the complement of an event $E$ . Let $E, F, G$ be pairwise independent events with $P (G) > 0$ and $P (E \cap F \cap G) = 0$ . Then $P (E^{c} \cap F^{c} ∣ G)$ equals

(A)

P (E^{c}) + P (F^{c})

(B)

P (E^{c}) - P (F^{c})

(C)

P (E^{c}) - P (F)

(D)

P (E) - P (F^{c})

JEE Advanced 1998

LEVELBoard

If $\overset{ˉ}{E}$ and $\overset{ˉ}{F}$ are the complementary events of events $E$ and $F$ respectively and if $0 < P (F) < 1$ , then

* Multiple Correct Options

(A)

P (E / F) + P (\overset{ˉ}{E} / F) = 1

(B)

P (E / F) + P (E / \overset{ˉ}{F}) = 1

(C)

P (\overset{ˉ}{E} / F) + P (E / \overset{ˉ}{F}) = 1

(D)

P (E / \overset{ˉ}{F}) + P (\overset{ˉ}{E} / \overset{ˉ}{F}) = 1

JEE Advanced 1982

LEVELBoard

If $A$ and $B$ are two events such that $P (A) > 0$ , and $P (B) \neq = 1$ , then $P (\frac{A ˉ}{B ˉ})$ is equal to

(A)

1 - P (\frac{A}{B})

(B)

1 - P (\frac{A ˉ}{B})

(C)

\frac{1 - P ( A \cup B )}{P ( B ˉ )}

(D)

\frac{P ( A ˉ )}{P ( B ˉ )}

JEE Advanced 2003

LEVELJEE Main

$A$ is targeting to $B, B$ and $C$ are targeting to $A$ . Probability of hitting the target by $A, B$ and $C$ are $2/3, 1/2$ and $1/3$ respectively. If $A$ is hit then find the probability that $B$ hits the target and $C$ does not.

JEE Advanced 1994

LEVELJEE Main

If two events $A$ and $B$ are such that $P (A^{c}) = 0.3, P (B) = 0.4$ and $P (A \cap B^{c}) = 0.5$ , then $P (B ∣ A \cup B^{c}) = .........$

JEE Advanced 1989

LEVELJEE Main

A pair of fair dice is rolled together till a sum of either 5 or 7 is obtained. Then the probability that 5 comes before 7 is .........

JEE Advanced 1985

LEVELJEE Main

A box contains 100 tickets numbered $1, 2, \dots, 100$ . Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than 10. The minimum number on them is 5 with probability .........

JEE Advanced 1981

LEVELBoard

For a biased die the probabilities for the different faces to turn up are given below : $Face Pr ob. 1 0.1 2 0.32 3 0.21 4 0.15 5 0.05 6 0.17$ This die is tossed and you are told that either face 1 or face 2 has turned up. Then the probability that it is face 1 is .........

JEE Main 2018 (16 April Shift 1)

LEVELJEE Main

Let A, B and C be three events, which are pair-wise independent and $\overset{ˉ}{E}$ denotes the complement of an event E. If $P (A \cap B \cap C) = 0$ and $P (C) > 0$ , then $P [(\overset{ˉ}{A} \cap \overset{ˉ}{B}) ∣ C]$ is equal to :

(A)

P (A) + P (\overset{ˉ}{B})

(B)

P (\overset{ˉ}{A}) - P (\overset{ˉ}{B})

(C)

P (\overset{ˉ}{A}) - P (B)

(D)

P (\overset{ˉ}{A}) + P (B)

JEE Main 2019 (11 January)

LEVELJEE Main

Two integers are selected at random from the set ${1, 2, \dots, 11}$ . Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

(A)

\frac{2}{5}

(B)

\frac{1}{2}

(C)

\frac{3}{5}

(D)

\frac{7}{10}

JEE Main 2019 (8 April Shift 1)

LEVELJEE Main

Let $A$ and $B$ be two non-null events such that $A \subset B$ . Then, which of the following statements is always correct ?

(A)

P (A ∣ B) = 1

(B)

P (A ∣ B) = P (B) - P (A)

(C)

P (A ∣ B) \leq P (A)

(D)

P (A ∣ B) \geq P (A)

JEE Main 2019 (10 April Shift 1)

LEVELJEE Main

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :

JEE Main 2020 (8 January Shift 1)

LEVELJEE Main

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 ?

(A)

P (A / (A \cup B)) = \frac{1}{4}

(B)

P (A / B^{'}) = \frac{1}{3}

(C)

P (A / B) = \frac{2}{3}

(D)

P (A^{'} / B^{'}) = \frac{1}{3}

JEE Main 2020 - 8 Jan (Morning)

LEVELBoard

Let $A$ and $B$ be two independent events such that $P (A) = 1/3$ and $P (B) = 1/6$ . Then, which of the following is TRUE?

JEE Main 2020 - 2 Sep (Evening)

LEVELJEE Main

Let $E^{C}$ denote the complement of an event $E$ . Let $E_{1}, E_{2}$ and $E_{3}$ be any pairwise independent events with $P (E_{1}) > 0$ and $P (E_{1} \cap E_{2} \cap E_{3}) = 0$ . Then $P (E_{2}^{C} \cap E_{3}^{C} / E_{1})$ is equal to :

JEE Main 2020 - 3 Sep (Morning)

LEVELJEE Main

A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared atleast once is :

JEE Main 2022 (24 June Shift 2)

LEVELBoard

A random variable $X$ has the following probability distribution: $X P (X) 0 k 1 2 k 2 4 k 3 6 k 4 8 k$ The value of $P (1 < X < 4∣ X \leq 2)$ is equal to:

(A)

\frac{4}{7}

(B)

\frac{2}{3}

(C)

\frac{3}{7}

(D)

\frac{4}{5}

JEE Main 2022 (25 June Shift 1)

LEVELJEE Main

Let $E_{1}$ and $E_{2}$ be two events such that the conditional probabilities $P (E_{1} ∣ E_{2}) = \frac{1}{2}$ , $P (E_{2} ∣ E_{1}) = \frac{3}{4}$ and $P (E_{1} \cap E_{2}) = \frac{1}{8}$ . Then:

(A)

P (E_{1} \cap E_{2}) = P (E_{1}) \cdot P (E_{2})

(B)

P (E_{1}^{'} \cap E_{2}^{'}) = P (E_{1}^{'}) \cdot P (E_{2})

(C)

P (E_{1} \cap E_{2}^{'}) = P (E_{1}) \cdot P (E_{2})

(D)

P (E_{1}^{'} \cap E_{2}) = P (E_{1}) \cdot P (E_{2})

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Three numbers are chosen at random without replacement from {1,2,3,...,8}. The probability that their minimum is 3, given that their maximum is 6, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If C and D are two events such that C⊂D and P(D)=0, then the correct statement among the following is

One ticket is selected at random from 50 tickets numbered 00, 01, 02, ..., 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } It is given that the events A and B are such that P(A)=1/4,P(A∣B)=1/2 and P(B∣A)=2/3. Then P(B) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let X and Y be two events such that P(X∣Y)=1/2,P(Y∣X)=1/3 and P(X∩Y)=1/6. Which of the following is (are) correct?

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Ec denote the complement of an event E. Let E,F,G be pairwise independent events with P(G)>0 and P(E∩F∩G)=0. Then P(Ec∩Fc∣G) equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If Eˉ and Fˉ are the complementary events of events E and F respectively and if 0<P(F)<1, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A and B are two events such that P(A)>0, and P(B)=1, then P(BˉAˉ​) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If two events A and B are such that P(Ac)=0.3,P(B)=0.4 and P(A∩Bc)=0.5, then P(B∣A∪Bc)=.........

A pair of fair dice is rolled together till a sum of either 5 or 7 is obtained. Then the probability that 5 comes before 7 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two integers are selected at random from the set {1,2,…,11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A and B be two non-null events such that A⊂B. Then, which of the following statements is always correct ?

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A and B be two independent events such that P(A)=31​ and P(B)=61​. Then, which of the following is TRUE ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A and B be two independent events such that P(A)=1/3 and P(B)=1/6. Then, which of the following is TRUE?

A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared atleast once is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A random variable X has the following probability distribution: XP(X)​0k​12k​24k​36k​48k​​ The value of P(1<X<4∣X≤2) is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let E1​ and E2​ be two events such that the conditional probabilities P(E1​∣E2​)=21​, P(E2​∣E1​)=43​ and P(E1​∩E2​)=81​. Then:

Three numbers are chosen at random without replacement from ${1, 2, 3, ..., 8}$ . The probability that their minimum is 3, given that their maximum is 6, is

If $C$ and $D$ are two events such that $C \subset D$ and $P (D) \neq = 0$ , then the correct statement among the following is

It is given that the events $A$ and $B$ are such that $P (A) = 1/4, P (A ∣ B) = 1/2$ and $P (B ∣ A) = 2/3$ . Then $P (B)$ is

Let $X$ and $Y$ be two events such that $P (X ∣ Y) = 1/2, P (Y ∣ X) = 1/3$ and $P (X \cap Y) = 1/6$ . Which of the following is (are) correct?

Let $E^{c}$ denote the complement of an event $E$ . Let $E, F, G$ be pairwise independent events with $P (G) > 0$ and $P (E \cap F \cap G) = 0$ . Then $P (E^{c} \cap F^{c} ∣ G)$ equals

If $\overset{ˉ}{E}$ and $\overset{ˉ}{F}$ are the complementary events of events $E$ and $F$ respectively and if $0 < P (F) < 1$ , then

If $A$ and $B$ are two events such that $P (A) > 0$ , and $P (B) \neq = 1$ , then $P (\frac{A ˉ}{B ˉ})$ is equal to

If two events $A$ and $B$ are such that $P (A^{c}) = 0.3, P (B) = 0.4$ and $P (A \cap B^{c}) = 0.5$ , then $P (B ∣ A \cup B^{c}) = .........$

Two integers are selected at random from the set ${1, 2, \dots, 11}$ . Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

Let $A$ and $B$ be two non-null events such that $A \subset B$ . Then, which of the following statements is always correct ?

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 ?

Let $A$ and $B$ be two independent events such that $P (A) = 1/3$ and $P (B) = 1/6$ . Then, which of the following is TRUE?

A random variable $X$ has the following probability distribution: $X P (X) 0 k 1 2 k 2 4 k 3 6 k 4 8 k$ The value of $P (1 < X < 4∣ X \leq 2)$ is equal to:

Let $E_{1}$ and $E_{2}$ be two events such that the conditional probabilities $P (E_{1} ∣ E_{2}) = \frac{1}{2}$ , $P (E_{2} ∣ E_{1}) = \frac{3}{4}$ and $P (E_{1} \cap E_{2}) = \frac{1}{8}$ . Then: