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

Solution Steps

To solve this problem, calculate the probability of drawing a heart on the first draw, then calculate the conditional probability of drawing a second heart given that the first card drawn was a heart. Multiply these probabilities together to find the overall probability of drawing two hearts in succession without replacement.

The total number of cards in a deck is 52, and the number of hearts is 13. The probability of drawing a heart on the first draw is given by: \[ P(\text{First Heart}) = \frac{13}{52} = 0.25 \]

After drawing one heart, there are now 12 hearts left and 51 cards left in total. The conditional probability of drawing a heart on the second draw given that the first card drawn was a heart is: \[ P(\text{Second Heart} \mid \text{First Heart}) = \frac{12}{51} \approx 0.2353 \]

The overall probability of drawing two hearts in succession without replacement is the product of the probabilities calculated in Steps 1 and 2: \[ P(\text{Two Hearts}) = P(\text{First Heart}) \times P(\text{Second Heart} \mid \text{First Heart}) = 0.25 \times 0.2353 \approx 0.0588 \]

Final Answer