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Wall Street Quant · Puzzles & Problems · question 43 of 155

You draw two cards from a standard deck without replacement. What is the probability that both cards are hearts?

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In a standard deck, there are 52 cards, with 13 cards of each suit (Spades, Hearts, Diamonds, and Clubs). The probability of both cards being Hearts can be calculated as follows:

First, consider the probability of drawing a Heart in the first draw. There are 13 Hearts and 52 cards in total, so the probability is:

$P(\text{1st card is Heart}) = \frac{13}{52} = \frac{1}{4}$

After the first draw, there are now 12 Hearts and 51 cards remaining in the deck. Now, consider the probability of drawing another Heart on the second draw, given the first card was a Heart:

$P(\text{2nd card is Heart} \, |\, \text{1st card is Heart}) = \frac{12}{51}$

To calculate the probability of both events happening, we can multiply the probabilities:

$P(\text{Both cards are Hearts}) = P(\text{1st card is Heart}) \times \\ \qquad \quad \quad \quad P(\text{2nd card is Heart} \, |\, \text{1st card is Heart})$

$P(\text{Both cards are Hearts}) = \frac{1}{4} \times \frac{12}{51}$

$P(\text{Both cards are Hearts}) = \frac{12}{204} = \frac{1}{17}$

So the probability that both cards are Hearts is $\frac{1}{17} \approx 5.882 \%$.

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