Question

In: Statistics and Probability

b. Suppose P(A) = 1 4 and P(B) = 2 5 . (a) What is the...

b. Suppose P(A) = 1 4 and P(B) = 2 5 .

(a) What is the maximum value of P(A ∩ B)?

(b) What is the minimum value of P(A ∩ B)? (

c) What is the maximum value of P(A ∩ B0 )

(d) What is the minimum value of P(A ∩ B0 )

(e) What is the maximum value of P(A ∪ B)?

(f) What is the minimum value of P(A ∪ B)?

(g) What is the maximum value of P(A|B)?

(h) What is the minimum value of P(A|B)?

Solutions

Expert Solution

P(A) = 1/4

P(B) = 2/5

(a)

Maximum value of P(A ∩ B) is the least of P(A) and P(B).

Therefore, Maximum value of P(A ∩ B) = 1/4

(b)

Minimum value of P(A ∩ B) is 0. It occurs when A and B are disjoint.

(c)

Maximum value of P(A ∩ B⁰ ) is the least of P(A) and P(B⁰).

P(B⁰) = 1- P((B)

= 3/5

Therefore, Maximum value of P(A ∩ B⁰ ) = 1/4

(d)

Minimum value of P(A ∩ B⁰ ) is 0. It occurs when A and B⁰ are disjoint.

(e)

Maximum value of P(A ∪ B) is P(A) + P(B)

We have, P(A ∪ B) = P(A) +P(B) - P(A ∩ B)

This is maximum when P(A ∩ B) is minimum, i.e when P(A ∩ B) is 0

Therefore, Maximum value of P(A ∪ B) = 1/4 + 2/5

= 13/20

(f)

Minimum value of P(A ∪ B) is the highest of P(A) and P(B)

Therefore, minimum value of P(A ∪ B) = 2/5

(g)

Maximum value of P(A|B) :

P(A|B) = P(A ∩ B)/P(B)

This is maximum when P(A ∩ B) is maximum

Therefore, Maximum value of P(A|B) = (1/4)/(2/5)

= 5/8

(h)

Minimum value of P(A|B) :

P(A|B) = P(A ∩ B)/P(B)

This is minimum when P(A ∩ B) is minimum

Therefore, minimum value of P(A|B) = 0

orchestra answered 1 year ago

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