Question

Use the verb diagram to calculate conditional probabilities

Which conditional probabilities are correct? Check all that apply.

P(D|F)=6/34
P(E|D)=7/25
P(D|E)=7/25
P(F|E)=8/18
P(E|F)=13/21

Answer 1

The Venn diagram explicitly tells us that the universe consists of 60 objects, and

`P(D\cap E\cap F)=\frac1{60}`

`P(D\cap E)=(1+6)/(60)=\frac7{60}`

`P(D\cap F)=(1+5)/(60)=\frac1{10}`

`P(E\cap F)=(1+7)/(60)=\frac2{15}`

`P(D)=(13+6+5+1)/(60)=\frac5{12}`

`P(E)=(4+6+7+1)/(60)=\frac3{10}`

`P(F)=(21+5+7+1)/(60)=(17)/(30)`

`P(D\cup E\cup F)=1-\frac3{60}=(19)/(20)`

By definition of conditional probability, we have

`P(D\mid F)=(P(D\cap F))/(P(F))=\frac{\frac1{10}}{(17)/(30)}=\frac3{17}`

`P(E\mid D)=\frac{\frac7{60}}{\frac5{12}}=\frac7{25}` (only this one is correct)

`P(D\mid E)=\frac{\frac7{60}}{\frac3{10}}=\frac7{18}`

`P(F\mid E)=\frac{\frac2{15}}{\frac3{10}}=\frac49`

`P(E\mid F)=\frac{\frac2{15}}{(17)/(30)}=\frac4{17}`