Question

Let S be a sample space and E and F be events associated with S. Suppose that Pr (Upper E )equals 0.6​, Pr (Upper F )equals 0.2 and Pr (Upper E intersect Upper F )equals 0.1. Calculate the following probabilities. ​(a) Pr (E|F )​(b) Pr (F|E )​(c) Pr (E| Upper F prime )​(d) Pr (Upper E prime | Upper F prime )

Answer 1

(a)0.5

(b)0.17

(c)0.625

(b)0.375

Explanation:

Pr(E)=0.6​

Pr(F)=0.2

`Pr(E\cap F)=0.1.`

(a)Pr (E|F )

`Pr (E|F )=(Pr(E \cap F))/(Pr(F)) \\=(0.1)/(0.2)\\\\=0.5`

(b)Pr (F|E )

`Pr (F|E )=(Pr(E \cap F))/(Pr(E)) \\=(0.1)/(0.6)\\\\=0.17`

(c)Pr (E|F')​

Pr(F')=1-P(F)

=1-0.2=0.8

`Pr(E \cap F')=P(E)-P(E\cap F)\\=0.6-0.1\\=0.5`

Therefore:

`Pr (E|F' )=(Pr(E \cap F'))/(Pr(F')) \\=(0.5)/(0.8)\\\\=0.625`

(d)Pr(E'|F')​

`P(E'\cap F')=P(E \cup F)'\\=1-P(E \cup F)\\=1-[P(E)+P(F)-P(E\cap F)]\\=1-[0.6+0.2-0.1]\\=1-0.7\\=0.3`

Therefore:

`Pr (E'|F' )=(Pr(E' \cap F'))/(Pr(F')) \\=(0.3)/(0.8)\\\\=0.375`