A pack of playing cards was found to contain only 51 cards. If the first 13 cards, which are examined are all red, then the probability that the missing card is black is :-

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`