Given that the events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5, and P(B) = p. Find p if they are(i) mutually exclusive

Given that the events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5, and P(B) = p. Find p if they are

(i) mutually exclusive

(ii) independent

Solution:

Given, P(A) = 1/2 ,

P (A ∪ B) = 3/5

and P(B) = p.

(1) For Mutually Exclusive

Given that, the sets A and B are mutually exclusive.

Thus, they do not have any common elements

Therefore, P(A ∩ B) = 0

We know that P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Substitute the formulas in the above-given formula, we get

3/5 = (1/2) + p – 0

Simplify the expression, we get

(3/5) – (1/2) = p

(6 − 5)/10 = p

1/10 = p

Therefore, p = 1/10

Hence, the value of p is 1/10, if they are mutually exclusive.

(ii) For Independent events:

If the two events A & B are independent,

we can write it as P(A ∩ B) = P(A) P(B)

Substitute the values,

= (1/2) × p

= p/2

Now, P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Now, substitute the values in the formula,

(3/5) = (1/2)+ p – (p/2)

(3/2)– (1/2)= p – (p/2)

(6 − 5)/10 = p/2

1/10 = p/2

p= 2/10

P = 1/5

Thus, the value of p is 1/5, if they are independent.