Algebra

Probability

Medium

core

For any events A and B of a sample space S, which of the following statements are TRUE?

(A) $P(S | B) = 1$

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

(C) $P(\bar{A} | B) = 1 - P(A | B)$

(D) $P(A | B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0$

Choose the correct answer from the options given below:

(A) and (B) only

(A), (C) and (D) only

(B) and (C) only

✅ Correct Option: 2

Statement (A): $P(S | B) = 1$

Using the conditional probability formula:

$P(S | B) = \frac{P(S \cap B)}{P(B)}$

Since $B \subseteq S$ , we have $S \cap B = B$

$P(S | B) = \frac{P(B)}{P(B)} = 1$

Statement (A) is TRUE.

Statement (B): $P(A \cap B) = P(A) + P(B) + P(A \cup B)$

The addition rule for probability states:

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

Rearranging:

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

The given statement has a plus sign instead of a minus sign.

Statement (B) is FALSE.

Statement (C): $P(\bar{A} | B) = 1 - P(A | B)$

Given that event $B$ has occurred, either $A$ occurs or $\bar{A}$ occurs:

$P(A | B) + P(\bar{A} | B) = 1$

Therefore:

$P(\bar{A} | B) = 1 - P(A | B)$

Statement (C) is TRUE.

Statement (D): $P(A | B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0$

This is the definition of conditional probability.

Statement (D) is TRUE.

Statements (A), (C), and (D) are TRUE.

Statement (B) is FALSE.

The correct answer is: (A), (C) and (D) only.

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2022: 25 May Shift 1

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[Given that $e^{-2} \approx 0.13534$ ]

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Let E and F are two independent events such that $P(E)=0.35$ and $P(E\cup F)=0.60$ then which of following statements are TRUE?

(A) $P(F)=\frac{5}{13}$

(B) $P(E|\overline{F})=0.35$

(D) $P(\overline{E}|\overline{F})=0.65$

Choose the correct answer from the options given below:

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A fair coin is tossed 100 times. The probability of getting head an odd number of times is

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