13. Compute the probability of drawing 4 hearts from a well-shuffled deck of 52 cards if the cards are not replaced after being drawn.

To compute the probability of drawing 4 hearts from a well-shuffled deck of 52 cards without replacement, we can use the concept of combinations. The probability of drawing the first heart is \( \frac{13}{52} \) since there are 13 hearts in a deck of 52 cards. After drawing the first heart, there are 12 hearts left in the deck of 51 cards. So, the probability of drawing the second heart is \( \frac{12}{51} \). Following this pattern, the probability of drawing the third heart is \( \frac{11}{50} \) and the probability of drawing the fourth heart is \( \frac{10}{49} \). To find the overall probability of drawing 4 hearts, we multiply these probabilities together: \[ P(\text{4 hearts}) = \frac{13}{52} \times \frac{12}{51} \times \frac{11}{50} \times \frac{10}{49} \] Let's calculate this probability. Calculate the value by following steps: - step0: Calculate: \(\frac{\frac{\frac{\frac{13}{52}\times 12}{51}\times 11}{50}\times 10}{49}\) - step1: Reduce the fraction: \(\frac{\frac{\frac{\frac{1}{4}\times 12}{51}\times 11}{50}\times 10}{49}\) - step2: Reduce the fraction: \(\frac{\frac{\frac{1}{17}\times 11}{50}\times 10}{49}\) - step3: Multiply: \(\frac{\frac{\frac{11}{17}}{50}\times 10}{49}\) - step4: Divide the terms: \(\frac{\frac{11}{850}\times 10}{49}\) - step5: Reduce the numbers: \(\frac{\frac{11}{85}}{49}\) - step6: Multiply by the reciprocal: \(\frac{11}{85}\times \frac{1}{49}\) - step7: Multiply the fractions: \(\frac{11}{85\times 49}\) - step8: Multiply: \(\frac{11}{4165}\) The probability of drawing 4 hearts from a well-shuffled deck of 52 cards without replacement is approximately 0.002641 or \( \frac{11}{4165} \).