Problem 1146

Two cards are drawn without replacement from an ordinary deck. Find the probability that two aces are drawn.




Notice that the problem statement can be written as P(E∩ F), where E and F represent the two events. The product rule of probability says that P(E ∩ F) = P(E).P(F|E).

First find P(E), the probability that the first card drawn is an ace.

P(E) = \frac{4}{52}=\frac{1}{13}

To find P(F|E) in this case, find the probability of drawing ace for the second card. Notice that there are 51 cards left in the deck an ace is drawn.

This partial deck of 51 cards is the reduced sample space used to calculate the probability when the second card is drawn.

After the first card is drawn from the deck, there are 3 aces left in the deck.

Thus, P(F|E), the probability that the second card chosen is an ace given the first card is an ace is \frac{3}{51} , or \frac{1}{17}.

Finally, find P(E∩F).

P(E∩F) = P(E).P(F|E)


Therefore, the probability that two aces are drawn is \frac{1}{221}.



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