Question

Given: p(e) = 0.42, p(f) = 0.51, and p(e ∪ f) = 0.70 part

a.find p(e ∩ f). part
b.find p(e | f). part
c.find p(f | e). part
d.are the events e and f independent?

Answer 1

`\mathbb P(E\cap F)=\mathbb P(E)+\mathbb P(F)-\mathbb P(E\cup F)=0.42+0.51-0.70=0.23`


`\mathbb P(E\mid F)=(\mathbb P(E\cap F))/(\mathbb P(F))=(0.23)/(0.51)\approx0.45`


`\mathbb P(F\mid E)=(\mathbb P(F\cap E))/(\mathbb P(E))=(0.23)/(0.42)\approx0.55`


`E` and
`F` are not independent. Independence would require that
`\mathbb P(E\cap F)=\mathbb P(E)*\mathbb P(F)`, but
`0.42*0.51\approx0.27\\eq0.23`.