Question

Let S be a sample space and E and F be events associated with S. Suppose that

Pr left parenthesis Upper E right parenthesis equals 0.6Pr(E)=0.6​,

Pr left parenthesis Upper F right parenthesis equals 0.2Pr(F)=0.2

and

Pr left parenthesis Upper E intersect Upper F right parenthesis equals 0.1Pr(E∩F)=0.1.

Calculate the following probabilities.

​(a) Pr left parenthesis E|F right parenthesisPr(E|F)	​(b) Pr left parenthesis F|E right parenthesisPr(F|E)
​(c) Pr left parenthesis E| Upper F prime right parenthesisPrE|F′	​(d) Pr left parenthesis Upper E prime | Upper F prime right parenthesisPrE′|F′

Answer 1

Solution :

Given that,

P(E) = 0.6

P(F) = 0.2 and P(F') = 1 - P(F) = 1 - 0.2 = 0.8

P(E F) = 0.1

P(E F) = P(E) + P(F) - P(E F) = 0.6 + 0.2 - 0.1 = 0.7

(a)

P(E | F) = P(E F) / P(F) = 0.1 / 0.2 = 0.5

(b)

P(F | E) = P(E F) / P(E) = 0.1 / 0.6 = 0.1667

(c)

P(E | F') = P(E F') / P(F') = [P(E) - P(E F)] /  P(F')

= [0.6 - 0.1] / 0.8

= 0.5 / 0.8

= 0.625

P(E | F') = 0.625

(d)

P(E' | F') = P(E' F') / P(F')

= [1 - P(E F)] / 0.8

= [1 - 0.7] / 0.8

= 0.3 / 0.8

= 0.375

P(E' | F') = 0.375