Question

ou find a deck of cards and know that the deck has too many clubs and spades in it. The deck has 59 cards. If you were to draw two cards without replacement, the probability of drawing a diamond and a club is 117 1711 . How many extra spades are in the deck?

Answer 1

We can find the number of extra clubs in the deck from the given probability, since the probability of drawing a diamond and a club is

$\dfrac{\binom{13}1\binom{13+c}1}{\binom{59}2}=\dfrac{13(13+c)}{1711}$

but this would imply that $c=-4$, which suggests we're taking 4 clubs out of the original deck and contradicts the problem statement that there are "too many clubs".

Perhaps the question is providing the probability of drawing a diamond, THEN a club, or vice versa. In that case, we have

$\dfrac{\binom{13}1}{\binom{59}1}\cdot\dfrac{\binom{13+c}1}{\binom{58}1}=\dfrac{13(13+c)}{3422}$

and solving gives $c=5$, which makes more sense. Then the number of extra spades in the deck must be 2.