Question

One of a pack of 52 cards has been lost; from the remainder of the pack two cards are drawn and are found to be spades; the chance that the missing card is a spade is $\frac{1}{5}+\frac{k}{50}$. Find the value of $k$.

Answer 1

The antecedent chance that the lost card is a spade is $\frac{13}{52}=\frac{1}{4},$ and the chance that is not a spade is $\frac{3}{4}.$
Thus $P_1=\frac{1}{4},P_2=\frac{3}{4}$
Also $P_1=\frac{12}{51}\times \frac{11}{50},P_2=\frac{13}{51}\times \frac{12}{50}$
$\therefore \frac{Q_1}{p_1{P}_1}=\frac{Q_2}{p_2{P}_2}=\frac{(Q_1+Q_2=1)}{p_1{P}_1+p_2{P}_2}$ yields
$\frac{Q_1}{11}=\frac{Q_2}{3\times 13}=\frac{1}{50}$
$\therefore Q_1=\frac{11}{50}=$ the chance that the missing card was a spade.