Question

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 are found to be all spades. Find the probability of the lost card being a spade.

Answer 1

Let $E_1=$ Event that lost card is a spade.

$E_2=$ Event that lost card is a non-spade.

$A=$ Event that three spades are drawn without replacement from 51 cards.

$P\left(E_1\right)=\frac{13}{52}=\frac{1}{4}$

$P\left(E_2\right)==1-\frac{1}{4}=\frac{3}{4}$

$P\left(\frac{A}{E_1}\right)=\frac{{}^{12}{C}_3}{{}^{51}{C}_3},P\left(\frac{A}{E_2}\right)=\frac{{}^{13}{C}_3}{{}^{51}{C}_3}$

$P\left(\frac{E_1}{A}\right)=\frac{\frac{1}{4}.\frac{{}^{12}{C}_3}{{}^{51}{C}_3}}{\frac{1}{4}.\frac{{}^{12}{C}_3}{{}^{51}{C}_3}+\frac{3}{4}.\frac{{}^{13}{C}_3}{{}^{51}{C}_3}}$

$=\frac{10}{49}$