Multiple choice

5 cards are drawn at random from a well shuffled pack of 52 playing cards. If it is known that there will be at least 3 hearts, the probability that there are 4 hearts is

  1. $\displaystyle \frac{^{13}C_{4}}{^{13}C_{3}+^{13}C_{4}+^{13}C_{5}}$
  2. $\displaystyle \frac{^{13}C_{4}}{^{13}C_{3}\times ^{39}C_{2}+^{13}c_{4}\times ^{39}C_{1}+^{13}C_{5}}$
  3. $\displaystyle \frac{^{13}C_{4}\times ^{39}C_{1}}{^{13}C_{3}\times ^{39}C_{2}+^{13}C_{4}\times ^{39}C_{1}+^{13}C_{5}}$
  4. $\displaystyle \frac{^{13}C_{4}}{^{13}C_{3}\times ^{13}C_{4}\times ^{13}C_{5}}$
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C Correct answer
Explanation

This is a conditional probability problem. Total ways to get at least 3 hearts is (13C3 * 39C2) + (13C4 * 39C1) + (13C5 * 39C0). The favorable outcome is getting exactly 4 hearts: (13C4 * 39C1). The ratio is the correct probability.

AI explanation

This requires the conditional probability of drawing exactly 4 hearts given that at least 3 hearts are drawn. The number of ways to draw exactly 4 hearts is 13C4 * 39C1, while the number of ways to draw at least 3 hearts is the sum of the ways to draw exactly 3, exactly 4, and exactly 5 hearts, which is 13C3 * 39C2 + 13C4 * 39C1 + 13C5. The probability is the ratio of the favorable outcomes to the total conditional outcomes, giving (13C4 * 39C1) / (13C3 * 39C2 + 13C4 * 39C1 + 13C5).

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