Hearts

Solution & Explanation

Part 1: Probability of Drawing a Heart

Scenario:

• A standard deck of 52 playing cards includes 13 hearts.
• One heart has been drawn, leaving 51 cards.
• We need to find the probability that the next card drawn is also a heart.

Solution:

• After one heart is drawn, there are 12 hearts remaining in the deck of 51 cards.
• The probability $P$ of drawing another heart is given by:

$P(\text{drawing a heart}) = \frac{\text{Number of remaining hearts}}{\text{Total remaining cards}} = \frac{12}{51}$

Explanation:

• Initially, there are 13 hearts in a full deck. Drawing one leaves 12 hearts.
• The probability is computed as the ratio of remaining hearts to the total remaining cards.

Part 2: Expected Number of Hearts in 13 Draws

Scenario:

• From a deck of 52 cards, 13 cards are drawn, including exactly 5 hearts.
• You then draw 13 cards from the remaining 39 cards.
• We need to calculate the expected number of hearts in your 13-card hand.

Solution:

• After the initial draw, there are 39 cards left, with 8 being hearts.
• The expected number of hearts $E[x]$ in your hand can be calculated using the hypergeometric distribution formula:

$E[x] = n \cdot \frac{K}{N}$

Where:

• $n = 13$ (number of draws)
• $K = 8$ (number of hearts remaining)
• $N = 39$ (total remaining cards)

Substituting these values:

$E[x] = 13 \cdot \frac{8}{39} = \frac{104}{39} \approx 2.67$

Explanation:

• The hypergeometric distribution is used to model scenarios where draws are made without replacement.
• The expected value formula $E[x] = n \cdot \frac{K}{N}$ gives the average number of successes (hearts) in $n$ draws.
• In this scenario, you can expect to draw approximately 2.67 hearts from the remaining deck.

Conclusion

• The probability of drawing a heart after one has already been drawn is $\frac{12}{51}$.
• The expected number of hearts in your hand when drawing 13 cards from the remaining deck is approximately 2.67.