91Ó°ÊÓ

There are 52 cards in a well-shuffled deck, which includes 13 diamond cards and 39 non-diamond cards. We will be drawing three cards without replacement.

To find the probability of drawing a diamond on the third draw, we first need to analyze different scenarios. There are three scenarios: 1. The first and second cards drawn are not diamonds: In this scenario, we have 39 non-diamond cards and 13 diamond cards remaining. 2. The first card is a diamond and the second card is not: In this scenario, we have 38 non-diamond cards and 12 diamond cards remaining. 3. The first card is not a diamond, but the second card is: In this scenario, we have 38 non-diamond cards and 12 diamond cards remaining. Notice that scenario 2 and 3 are identical; hence, we can consider the joint scenario and calculate the probability of drawing a diamond on the third draw.

1. Probability of drawing non-diamond, non-diamond cards, and then a diamond card: First two non-diamond cards: \( \frac{39}{52} \times \frac{38}{51} \) The third card is a diamond: \( \frac{13}{50} \) Total probability for this scenario: \( \frac{39}{52} \times \frac{38}{51} \times \frac{13}{50} \) 2. Probability of (drawing a diamond and then a non-diamond) or (drawing a non-diamond and then a diamond), and then drawing a diamond card: First card: diamond; second card: non-diamond: \( \frac{13}{52} \times \frac{39}{51} \) First card: non-diamond; second card: diamond: \( \frac{39}{52} \times \frac{13}{51} \) Total probability for this scenario: We need to multiply each case probability by \(\frac{12}{50}\), which is the probability of drawing a diamond card as the third card. \( \frac{12}{50} \times \left(\frac{13}{52} \times \frac{39}{51} + \frac{39}{52} \times \frac{13}{51} \right) \)

Add the probabilities from step 3 to find the total probability. \( \frac{39}{52} \times \frac{38}{51} \times \frac{13}{50} + \frac{12}{50} \times \left(\frac{13}{52} \times \frac{39}{51} + \frac{39}{52} \times \frac{13}{51} \right) \)

Simplify the expression to get the final probability. \(=\frac{13}{50}\) The probability that the third card drawn is a diamond is \(\frac{13}{50}\) or 26%.