Algebra

Probability

Medium

core

Which of the following statements are correct?

(A) If E and F are independent events then $P(E \cap F) = P(E) \cdot P(F)$

(B) If E and F are mutually exclusive events, then $P(E \cup F) = P(E) + P(F) - P(E) \cdot P(F)$

(C) The conditional probability of an event E, given the occurrence of the event F is given by $\frac{P(E \cap F)}{P(F)}, P(F) \neq 0$

(D) If E and F be the events associated with the sample space S of an experiment, then $P(\overline{E}|F) = 2 - P(E|F)$

Choose the correct answer from the options given below:

(A) and (C) only

(A), (B) and (C) only

(B) and (D) only

✅ Correct Option: 1

Statement (A): If E and F are independent events, then $P(E \cap F) = P(E) \cdot P(F)$

This is the definition of independent events — correct.

Statement (B): If E and F are mutually exclusive, then $P(E \cup F) = P(E) + P(F) - P(E) \cdot P(F)$

Mutually exclusive means $P(E \cap F) = 0$

Using the addition formula:

$P(E \cup F) = P(E) + P(F) - P(E \cap F)$

$= P(E) + P(F) - 0$

$= P(E) + P(F)$

The formula given in (B) is the union formula for independent events, not mutually exclusive events — incorrect.

Statement (C): $P(E|F) = \dfrac{P(E \cap F)}{P(F)},\; P(F) \neq 0$

This is the definition of conditional probability — correct.

Statement (D): $P(\overline{E}|F) = 2 - P(E|F)$

Since E and $\overline{E}$ together cover the entire sample space:

$P(E|F) + P(\overline{E}|F) = 1$

$P(\overline{E}|F) = 1 - P(E|F)$

The statement uses $2$ instead of $1$ — incorrect.

Statements (A) and (C) are correct.

2026: 25 May Shift 1

For two events A and B, if $P(A) = 0.6$ , $P(B) = 0.5$ and $P(A\cup B) = 0.8$ . Then which of the following statements are TRUE ?

A. $P(A\cap B) = 0.3$ .

B. A and B are independent events.

C. $P(A'\cap B') = 0.2$ .

D. A and B are mutually exclusive events.

Choose the correct answer from the options given below:

2025: 30 May Shift 2

For a random variable x, probability distribution P(x) is given by $P(x) = \frac{k}{6}(3-x), x = 0, 1, 2$ , then

Match List-I with List-II

List-I	List-II
(A) k is equal to	(i) $\frac{1}{2}$
(B) P(x = 0)	(ii) 1
(C) P(x < 2)	(iii) $\frac{1}{6}$
(D) P(1 < x ≤ 2)	(iv) $\frac{5}{6}$

Choose the correct answer from the options given below:

2024: 16 May Shift 1

Match the options of List-I to List-II :

List-I	List-II
(A) $k$	(I) $\frac{7}{10}$
(B) $P(X < 3)$	(II) $\frac{53}{100}$
(C) $P(X > 2)$	(III) $\frac{1}{10}$
(D) $P(2 < X <7 )$	(IV) $\frac{3}{10}$

Choose the correct answer from the options given below :

2026: 22 May Shift 1

Two dice are thrown simultaneously. If X denotes the number of sixes, then the mean of X is :

2026: 21 May Shift 1

A man is known to speak the truth 3 out of 4 times. He throw a die and reports that it is five then the probability that it is actually five is

2024: 16 May Shift 1

There are $6$ cards numbered $1$ to $6$ , one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two cards drawn. Then $\mathrm{P}(\mathrm{X}>3)$ is :

Solution

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