There are 8 black balls and 8 red balls in an urn. If 4 balls are drawn without replacement, what is the probability that exactly 1 black ball is drawn? Express your answer as a fraction or a decimal number rounded to four decimal places.
Answer
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Step 1 ：We are given that there are 8 black balls and 8 red balls in an urn. We are asked to find the probability of drawing exactly 1 black ball from 4 draws without replacement.

Step 2 ：This is a combination problem. The total number of ways to draw 4 balls from 16 (8 black and 8 red) is given by the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), where n is the total number of items, k is the number of items to choose, and '!' denotes factorial.

Step 3 ：The number of ways to draw 1 black ball from 8 is also given by the combination formula. The number of ways to draw the remaining 3 balls from the remaining 7 red balls is also given by the combination formula.

Step 4 ：The probability is then the number of successful outcomes (drawing 1 black and 3 red balls) divided by the total number of outcomes (drawing 4 balls from 16).

Step 5 ：Calculating the total number of ways to draw 4 balls from 16, we get 1820.

Step 6 ：Calculating the number of ways to draw 1 black ball from 8, we get 8.

Step 7 ：Calculating the number of ways to draw 3 red balls from the remaining 7, we get 56.

Step 8 ：Dividing the number of successful outcomes by the total number of outcomes, we get a probability of approximately 0.24615384615384617.

Step 9 ：Final Answer: The probability that exactly 1 black ball is drawn is approximately \(\boxed{0.2462}\).