If A and B are mutually exclusive events with \(\rm P(A)=\frac{1}{2} P(B)\), then P(A) = ?

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AAI ATC Junior Executive 21 Feb 2023 Shift 2 Official Paper
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  1. \(\frac{1}{3}\)
  2. \(\frac{1}{6}\)
  3. \(\frac{1}{4}\)
  4. \(\frac{1}{2}\)

Answer (Detailed Solution Below)

Option 1 : \(\frac{1}{3}\)
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Detailed Solution

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Given:

 A and B are mutually exclusive events with \(\rm P(A)=\frac{1}{2} P(B)\)

Concept:

If A and B are mutually exclusive events then

P(A ∩ B) = 0

Calculation:

A and B are mutually exclusive events then

P(A ∩ B) = 0

we know that 

P( A ⋃ B) = P(A) + P(B) - P(A ∩ B)

⇒ P( A ⋃ B) = P(A) + P(B)

On substituting the values P(A ⋃ B) means total probability then

⇒ 1 = P(A) + 2P(A) (since \(\rm P(A)=\frac{1}{2} P(B)\))

⇒ 3P(A) = 1

⇒ \(\rm P(A)=\frac{1}{3}\)

Hence the option (1) is correct.

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