Question

A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability 5/8, French with probability 5/8, and art and French together with probability 1/4. What is the probability that he chooses mathematics? What is the probability that he chooses either art or French?

Answer 1

The probability of choosing mathematics =P(M)=
`(3)/(4)`

The probability that he chooses either art or French=1

Explanation:

We are given that a student must choose exactly two out of three elective subjects : art ,french and mathematics.

The probability of choosing art=
`(5)/(8)`

The probability of choosing french =
`(5)/(8)`

The probability of choosing French and art=
`(1)/(4)`

Let A ,F and M denotes the students of art,french and mathematics.

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

`P(A\cap F)+P(A\cap M)=(5)/(8)`

`P(F\cap M)+P(F\cap A)=P(F)=(5)/(8)`

Probability of choosing mathematics only=0

Probability of choosing French only =0

Probability of choosing art only =0

Probability of choosing all three subjects =0

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

`P(A\cap F)=(1)/(4)`

Substitute the value then we get

`P(A\cap M)=(5)/(8)-(1)/(4)==(3)/(8)`

`P(F\cap M)=(5)/(8)-(1)/(4)=(3)/(8)`

Therefore,
`P(M)=P(A\cap M)+P(F\cap M)=(3)/(8)+(3)/(8)=(6)/(8)=(3)/(4)`

Hence, the probability of choosing mathematics =P(M)=
`(3)/(4)`

`P(A\cup F)=P(A)+P(F)-P(A\cap F)`

`P(A\cup F)=(5)/(8)+(5)/(8)-(1)/(4)`

`P(A\cup F)=(5+5-2)/(8)=1`

Hence, the probability that he chooses either art or French=1