Question

Compound events help!!!!

Answer 1

Let
`M,F,P` denote the events that a student is male, a student is female, and a student has a phone, respectively. Then we're given

`P(M)=(420)/(800)`

`P(F)=1-P(M)=(380)/(800)`

`P(P)=(325)/(800)`

We want to determine
`P(M\cup P)`, which is equivalent to

`P(M\cup P)=P(M)+P(P)-P(M\cap P)`

By the law of total probability, we have

`P(P)=P(M\cap P)+P(F\cap P)`

That is, any student belonging to
`P` will be either male or female, and the events
`M` and
`F` are mutually exclusive of one another (because they are complementary). We're told that

`P(F\mid P)=(200)/(325)`

By definition of conditional probability,

`P(F\mid P)=(P(F\cap P))/(P(P))=(P(F\cap P))/((325)/(800))\implies P(F\cap P)=(200)/(800)`

and so

`P(P)=P(M\cap P)+P(F\cap P)\iff(325)/(800)=P(M\cap P)+(200)/(800)\implies P(M\cap P)=(125)/(800)`

So we have

`P(M\cup P)=(420)/(800)+(325)/(800)-(125)/(800)=(620)/(800)`