A card from a pack of 52 cards is lost. From the remaining cards of the pack; two cards are drawn and are found to be hearts. Find the probability of the missing card to be a heart.

Answer

`11//50`

Explanation

Let `A_(1)` be the event that missing card is spade and `A_(2)` be event that missing card is non-spade. Then,
`P(A_(1))=1/4,P(A_(2))=3/4`
Let A be the event that 2 spade cards are drawn from the remaining cards. Then,
`P((A)/(A_(2)))=(""^(12)C_(2))/(""^(51)C_(2))andP((A)/(A_(2)))=(""^(13)C_(2))/(""^(51)C_(2))`
`impliesP(A)=P(A_(1))P((A)/(A_(1)))+P(A_(2))P((A)/(A_(2)))`
`=1/4(""^(12)C_(2))/(""^(51)C_(2))+3/4(""^(13)C_(2))/(""^(51)C_(2))`
`=(1)/(4xx""^(51)C_(2))[""^(12)C_(2)+3xx""^(13)C_(2)]`
Now, `P((A_(1))/(A))=(P(A_(1))P((A)/(A_(1))))/(P(A))`
`=((1)/(4)(""^(12)C_(2))/(""^(51)C_(2)))/((1)/(4xx""^(51)C_(2))[""^(12)C_(2)+3xx""^(13)C_(2)])=11/50`