What is the probability of drawing two hearts sequentially from a standard deck of 52 playing cards without replacement?

To find the probability of drawing two hearts sequentially from a standard deck of 52 playing cards without replacement, we start by analyzing the process step by step.

Initially, there are 52 cards in total, among which 13 cards are hearts. The probability of drawing the first heart is the ratio of the number of hearts to the total number of cards. Thus, the probability of drawing a heart first is:

[

\frac{13}{52} = \frac{1}{4}

]

Once the first heart is drawn, we have a total of 51 cards left in the deck, of which 12 are hearts (since one heart has already been drawn). The probability of then drawing a second heart is:

[

\frac{12}{51}

]

To find the total probability of both events happening—drawing two hearts in succession—the two probabilities must be multiplied together:

[

\text{Total Probability} = \frac{13}{52} \times \frac{12}{51}

]

Calculating this gives:

[

\frac{13 \times 12}{52 \times 51} = \frac{156}{2652}

]

Next, simplifying that fraction results