Find `k` if `fog=gof` where `f(x)=4x+8`, `g(x)=3x+k`.

Explanation

Let `A_(1)` be the event that black card is lot, `A_(2)` be the event that red card is lost, and let A denote occurrence of first 13 cards which are examined an d are found to be all red. Then, we have to find `P(A_(1)//A)."We have "P(A_(1))=P(A_(2))=1//2.Also P(A//A_(1))=""^(26)C_(13)//^(51)C_(13)and P(A//A_(2))=""^(25)C_(13)//^(51)C_(13).` Then by Bayes's rule,
`P(A_(1)//A)(P(A_(1))P(A//A_(1)))/(P(A_(1))P(A//A_(1))+P(A_(2))P(A//A_(2)))`
`=(1/2(""^(26)C_(13))/(""^(51)C_(13)))/((1)/(2)(""^(26)C_(13))/(""^(51)C_(13))+1/2(""^(25)C_(13))/(""^(51)C_(13)))`
`=(""^(26)C_(13))/(""^(26)C_(13)+""^(25)C_(13))=(2)/(2+1)=2/3`