If A and B be two events such that P (A) = 1/4, P (B) = 1/3 and P (A ∪ B) = 1/2, show that A and B are independent events.

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Solution

$P (A \cup B) = P (A) + P (B) - P (A \cap B) \Rightarrow P (A \cap B) = P (A) + P (B) - P (A \cup B) \Rightarrow P (A \cap B) = \frac{1}{4} + \frac{1}{3} - \frac{1}{2} \Rightarrow P (A \cap B) = \frac{3 + 4 - 6}{12} \Rightarrow P (A \cap B) = \frac{1}{12} = \frac{1}{4} \times \frac{1}{3} = P (A) P (B) Thus, A and B are independent events.$

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Similar questions

Q. If

A

and

B

are two independent events such that

P (¯ ¯¯ ¯ A \cap B) = \frac{2}{15}

and

P (A \cap ¯ ¯¯ ¯ B) = \frac{1}{6},

then find

P (A)

and

P (B) .

If A and B are two independent events such that $P (A \cap B) = \frac{1}{6}$ and $P (¯ ¯¯ ¯ A \cap ¯ ¯¯ ¯ B) = \frac{1}{3},$ then write the values of P(A) and P(B).

Q. Assertion :For two events A & B such that

P (A) = P (A / B) = \frac{1}{4}

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

then

P (¯ A / B) = \frac{3}{4}

P (¯ B / A) = \frac{1}{2}

. Reason: Two events A & B are independent then

P (B / A) = P (B)

Q. If A and B are two independent events such that P (

A

∩ B) = 2/15 and P (A ∩

B

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Q. If A and B are two independent events such that

P (¯ A \cap B) = \frac{2}{15}

and

P (A \cap ¯ B) = \frac{1}{6}

, then P(B) is

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If A and B be two events such that P (A) = 1/4, P (B) = 1/3 and P (A ∪ B) = 1/2, show that A and B are independent events.

PA∪B =PA+ PB-PA∩B⇒PA∩B =PA+ PB-PA∪B⇒PA∩B =14+13-12⇒PA∩B =3+4-612⇒PA∩B =112=14×13=PAPBThus, A and B are independent events.