A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and found to be all spades. Find the probability of the lost card being a spade.

Explanation

A=lost car is spade

`P(A)=1/4,P(a')=3/4`

B=3 cards are all spades

`P(A|B)=(P(B|A)P(A))/(P(B|A)P(A)+P(B|A')P(A)`

`P(B|A)` Now 51 cards,12 spades+39 not spade

`P(B|A)=12C_3/51C_3=1320/(51*50*49)`

`P(B|A')`,51 cards,13 spades+38 not spades

`P(B|A')=(13C_3)/(51C_3)=(13*132)/(51*50*49)`

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