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

Let E1: a spade is lost
A: two cards drawn are spade
Here $P\left ( E_{1} \right )= \frac{13}{52}= \frac{1}{4}\; \; \; \; P\left ( E_{2} \right )= 1-\frac{1}{4}= \frac{3}{4}$
$P\left ( \frac{A}{E_{1}} \right )= \frac{^{12}C_{2}}{^{51}C_{2}},P\left ( \frac{A}{E_{2}} \right )= \frac{^{13}C_{2}}{^{51}C_{2}}$
By baye's Theorem $= P\left (\frac{ E_{1}}{A} \right )= \frac{P\left ( E_{1} \right )P\left ( \frac{A}{ E_{1}} \right )}{P\left ( E_{1} \right )P\left ( \frac{A}{ E_{1}} \right )+P\left ( E_{2} \right )P\left ( \frac{A}{ E_{2}} \right )}$
$\Rightarrow P\left ( \frac{ E_{1}}{A} \right )= \frac{\frac{1}{4}\times \frac{22}{425}}{\frac{1}{4}\times \frac{22}{425}+\frac{3}{4}\times \frac{26}{425}}= \frac{22}{22+78}$
$\Rightarrow \therefore P\left ( \frac{E_{1}}{A} \right )= \frac{11}{50}$

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