91Ó°ÊÓ

Before diving into the calculations, let's analyze the problem we have. We need to draw cards without replacement, meaning each card removed from the deck will not be put back. Also, as we are calculating the probability, we will be using combinations to find the different ways to draw cards.

First, we are going to find the total number of possible combinations of drawing 6 cards from a deck of 52 cards. This can be represented as \( C(52, 6) \), which can be calculated using the formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] In our case, n = 52 and k = 6, and the total number of combinations will be: \[ C(52, 6) = \frac{52!}{6!(52-6)!} = 20,358,520 \] So there are 20,358,520 different ways to draw 6 cards from a deck of 52 cards.

Now, we'll find the favorable combinations, where the third spade is drawn on the sixth draw. We can break this down into two parts: the first 5 cards and the 6th card. 1. The first 5 cards must contain exactly 2 spades. There are 13 spades in the deck, so there are \( C(13, 2) \) ways to choose 2 spades. Out of the remaining 3 cards, we need to choose cards from the other 39 non-spade cards in the deck. There are \( C(39, 3) \) ways to do that. So the total number of ways to draw these 5 cards is: \[ C(13, 2) \times C(39, 3) = \frac{13!}{2!(13-2)!} \times \frac{39!}{3!(39-3)!} = 4,201,835 \] 2. The 6th card must be a spade. As there are already 2 spades in the first 5 cards, there are now 11 spades left in the remaining 47 cards. There are \( C(11, 1) \) ways to choose the 6th card as a spade: \[ C(11, 1) = \frac{11!}{1!(11-1)!} = 11 \] Now, we can calculate the total number of favorable combinations: \[ total\_favorable\_combinations = 4,201,835 \times 11 = 46,220,185 \]

Finally, we can find the probability of the third spade appearing on the sixth draw by dividing the number of favorable combinations by the total number of combinations: \[ probability = \frac{favorable\_combinations}{total\_combinations} = \frac{46,220,185}{20,358,520} \approx 0.2267 \] So the probability that the third spade appears on the sixth draw is approximately 0.2267 or 22.67%.