Question

P(a)=0.5 p(a|b)=25 p(b|a)=0.2 find p(b) and find p(not b|not a)

Answer 1

By the definition of conditional probability, you have


`\mathbb P(B|A)=(\mathbb P(B\cap A))/(\mathbb P(A))\implies\mathbb P(A\cap B)=0.2*0.5=0.1`


`\mathbb P(A|B)=(\mathbb P(A\cap B))/(\mathbb P(B))\implies\mathbb P(B)=(0.1)/(0.25)=1`


`\mathbb P(\\eg B|\\eg A)=(\mathbb P(\\eg B\cap\\eg A))/(\mathbb P(\\eg A))`

Recall that


`\mathbb P(\\eg B\cap\\eg A)=\mathbb P(\\eg(B\cup A))=1-\mathbb P(B\cup A)`

and that


`\mathbb P(B\cup A)=\mathbb P(B)+\mathbb P(A)-\mathbb P(B\cap A)`

This means


`\mathbb P(\\eg B|\\eg A)=(1-\mathbb P(B)-\mathbb P(A)+\mathbb P(B\cap A))/(1-\mathbb P(A))=(1-0.4-0.5+0.1)/(1-0.5)=0.1`