Solve the following:

The ratio of Boys to Girls in a college is 3:2 and 3 girls out of 500 and 2 boys out of 50 of that college are good singers. A good singer is chosen what is the probability that the chosen singer is a girl?

#### Solution

Let event S: Student is a good singer,

event B: Student is a boy,

event G: Student is a girl.

Since the ratio of boys to girls is 3:2 and

3 girls out of 500 and 2 boys out of 50 are good singers.

∴ P(B) = `3/5`, P(G) = `2/5`, `"P"("S"/"G") = 3/500`, `"P"("S"/"B") = 2/50`.

`"P"("S") = "P"("G") xx "P"("S"/"G") + "P"("B") xx "P"("S"/"B")`

= `2/5 xx 3/500 + 3/5 xx 2/50`

= `(2xx3)/5(1/500+1/50)`

= `6/5 xx 11/500`

= `33/1250`

Required probability = `"P"("G"/"S")`

By Bayes’ theorem,

`"P"("G"/"S") = ("P"("G") "P"("S"/"G"))/("P"("S")`

= `(2/5 xx 3/500)/(33/1250)`

= `1/11`