Given that the events A and B are such that and P (B) = p . Find p if they are (i) mutually exclusive (ii) independent.

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Solution

(i)

The probability of event A is 1 2 and the probability of event B is p and P( A∪B )= 3 5 .

The formula for two mutually exclusive events is,

P( A∪B )=P( A )+P( B )−P( A∩B )

Since, the events A and B are mutually exclusive so P( A∩B )=0.

Substitute the given values in the above formula we get,

3 5 = 1 2 +p−0 p= 3 5 − 1 2 p= 1 10

Thus, the value of P( B )= 1 10 .

(ii)

The formula for two independent events is,

P( A∩B )=P( A )⋅P( B )

Substitute the given values in the above formula we get,

P( A∩B )= 1 2 ×p = p 2

The formula for two mutually exclusive events is,

P( A∪B )=P( A )+P( B )−P( A∩B )

Substitute the given values in the above formula we get,

3 5 = 1 2 +p− p 2 3 5 − 1 2 = 2p−p 2 1 10 = p 2 p= 1 5

Thus, the value of P( B )= 1 5 .

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Similar questions

Given that the events A and B are such that P(A) = $\frac{1}{2}$ , $P (A \cup B) = \frac{3}{5}$ and P (B) = p. Find p if they are (i) mutually exclusive (ii) independent.

Given that the events A and B are such thatand P (B) = p. Find p if they are (i) mutually exclusive (ii) independent.

Q. Given that the events A and B are such that

P (A) = \frac{1}{2}, P (B) = p, P (A \cup B) = \frac{3}{5}

. Find p if A and B are. (I) mutually exclusive (ii) independent.

Q. Suppose

A

and

B

are two events with

P (A) = 0.5

and

P (A \cup B) = 0.8

. Let

P (B) = p

A

and

B

are mutually exclusive and

P (B) = q

A

and

B

are independent events, then the value of

q / p

Q. Assertion :If the independent events

A

B

are such that

0 < P (A) < 1, 0 < P (B) < 1

, then

A

B

can not be mutually exclusive. Reason: Two events

A

B

can not be mutually exclusive, if

P (A \cap B) \neq 0

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