Question

We’re going to pick a 12-person jury randomly from a group of 11 Communists and 15 dermatologists. What is the probability that the jury will be composed of exactly 5 Communists and exactly 7 dermatologists?

Answer 1

The probability that a jury composed of exactly 5 Communists and 7 dermatologists is calculated using the hypergeometric distribution formula considering combinations of selections from each group without replacement.

Step-by-step explanation:

The probability that the jury will be composed of exactly 5 Communists and exactly 7 dermatologists can be calculated using the hypergeometric distribution. This distribution is used when we are sampling from two distinct groups without replacement. In this case, we have two groups: Communists and dermatologists. We want to know the probability of selecting exactly 5 Communists out of 11 and 7 dermatologists out of 15 to form a 12-person jury.

To calculate this probability, we use the hypergeometric probability formula:
P(X=k) = [(C(group1, k) * C(group2, n-k)) / C(total, n)], where C represents the combination formula 'n choose k'.

For the 5 Communists, the number of ways to choose them is C(11, 5). For the 7 dermatologists, the number of ways to choose them is C(15, 7). The total ways to choose 12 people out of the whole group is C(26, 12). Plugging the numbers into the formula gives us:
P(X=5) = [C(11, 5) * C(15, 7) / C(26, 12)]

After calculating these combinations and plugging them into the formula, we get the exact probability.