If A and B are two independent events, then which of the following is/are true? (A) P(A B) = 0 (B) P(A B) = 1 - P(A')P(B') (C) P(A B) = P(A)P(B) (D) P(A B) = P(A)P(B) Choose the correct answer from the options given below: Option 1: (B) and (D) only Option 2: (A) and (C) only Option 3: (B) only Option 4: (B) and (C) only

Answer: (B) and (D) only. Solution: Two events A and B are independent when one event happening doesn't affect the probability of the other event happening. For independent events: P(A B) = P(A) x P(B) Statement (A): P(A B) = 0 For independent events: P(A B) = P(A) x P(B) This equals 0 only if P(A) = 0 or P(B) = 0. Independent events can occur together, so this statement is false. Statement (B): P(A B) = 1 - P(A')P(B') Using the complement rule: P(A B) = 1 - P(A B)' Applying De Morgan's Law: (A B)' = A' B' P(A B) = 1 - P(A' B') Since A and B are independent, A' and B' are also independent: P(A' B') = P(A') x P(B') Therefore: P(A B) = 1 - P(A')P(B') This statement is true. Statement (C): P(A B) = P(A)P(B) The correct formula for union is: P(A B) = P(A) + P(B) - P(A B) For independent events: P(A B) = P(A) + P(B) - P(A)P(B) This is not equal to P(A)P(B). This statement is false. Statement (D): P(A B) = P(A)P(B) This is the definition of independent events. When two events are independent, the probability of both happening together equals the product of their individual probabilities. This statement is true. Statements (B) and (D) are true.

Algebra

Probability

Medium

core

If A and B are two independent events, then which of the following is/are true?

(A) $P(A \cap B) = 0$

(B) $P(A \cup B) = 1 - P(A')P(B')$

(C) $P(A \cup B) = P(A)P(B)$

(D) $P(A \cap B) = P(A)P(B)$

Choose the correct answer from the options given below:

(B) and (D) only

(A) and (C) only

(B) only

(B) and (C) only

✅ Correct Option: 1

Two events A and B are independent when one event happening doesn't affect the probability of the other event happening.

For independent events: $P(A \cap B) = P(A) \times P(B)$

Statement (A): $P(A \cap B) = 0$

For independent events: $P(A \cap B) = P(A) \times P(B)$

This equals 0 only if $P(A) = 0$ or $P(B) = 0$ .

Independent events can occur together, so this statement is false.

Statement (B): $P(A \cup B) = 1 - P(A')P(B')$

Using the complement rule:

$P(A \cup B) = 1 - P(A \cup B)'$

Applying De Morgan's Law:

$(A \cup B)' = A' \cap B'$

$P(A \cup B) = 1 - P(A' \cap B')$

Since A and B are independent, A' and B' are also independent:

$P(A' \cap B') = P(A') \times P(B')$

Therefore:

$P(A \cup B) = 1 - P(A')P(B')$

This statement is true.

Statement (C): $P(A \cup B) = P(A)P(B)$

The correct formula for union is:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

For independent events:

$P(A \cup B) = P(A) + P(B) - P(A)P(B)$

This is not equal to $P(A)P(B)$ .

This statement is false.

Statement (D): $P(A \cap B) = P(A)P(B)$

This is the definition of independent events. When two events are independent, the probability of both happening together equals the product of their individual probabilities.

This statement is true.

Statements (B) and (D) are true.

2026: 18 May Shift 1

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2025: 2 June Shift 1

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Choose the correct answer from the options given below:

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2025: 2 June Shift 1

A problem in mathematics is given to three students whose chances of solving it are 1/2, 1/3, 1/4 respectively. The probability that the problem is solved is

Solution

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