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Q.
Two cards from an ordinary deck of $52$ cards are missing. What is the probability that a random card drawn from this deck is a spade?
Probability - Part 2
Solution:
Let $E$ be the event that the randomly drawn card is a spade. Let $F_i$ be the event that spades are missing from the $50$-card (defective) deck, for $i = 0$, $1$, $2$.
We want $P(E)$, which we compute by conditioning on how many spades are missing from the original deck :
$P(E) = P(E|F_0) P(F_0) + P(E|F_1) P(F_1) + P(E|E_2) P(F_2)$
$= \frac{13}{50} \frac{\binom{13}{0}\binom{39}{2}}{\binom{52}{2}}+\frac{12}{50} \frac{\binom{13}{1}\binom{39}{1}}{\binom{52}{2}}+\frac{11}{50} \frac{\binom{13}{2}\binom{39}{0}}{\binom{52}{2}}$
$ = \frac{1}{4}$