NIMCET 2013 — Mathematics PYQ

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Question

NIMCET 2013 — Mathematics PYQ

NIMCET | Mathematics | 2013

If $A$ and $B$ are two events such that $P(A \cup B) = \frac{5}{6}$ , $P(A \cap B) = \frac{1}{3}$ and $P(\bar{B}) = \frac{1}{2}$ , then the events $A$ and $B$ are:

Choose the correct answer:

Explanation

Solution

1. Find $P(B)$ :

Given $P(\bar{B}) = \frac{1}{2}$ .

P(B) = 1 - P(\bar{B}) = 1 - \frac{1}{2} = \frac{1}{2}

2. Find $P(A)$ using the Addition Theorem of Probability:

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

\frac{5}{6} = P(A) + \frac{1}{2} - \frac{1}{3}

\frac{5}{6} = P(A) + \frac{3 - 2}{6}

\frac{5}{6} = P(A) + \frac{1}{6}

P(A) = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}

3. Check for Independence:

Events are independent if $P(A \cap B) = P(A) \times P(B)$ .

P(A) \times P(B) = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3}

Given $P(A \cap B) = \frac{1}{3}$ .

Since $P(A \cap B) = P(A) \times P(B)$ , the events $A$ and $B$ are Independent.