Final answer:
The probability that four out of six people received the treatment and two received the placebo in a clinical trial is approximately 34.65%. This calculation uses combinations to determine the different ways participants could be chosen from the treatment and placebo groups, and then to find the overall probability.
Step-by-step explanation:
The probability that four people received the treatment and two received the placebo in a clinical trial group of six people can be calculated using combinations. First, we need to find the number of ways to choose four people out of 15 who received treatment, and two people out of 10 who received the placebo. This can be done using the combination formula C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items to choose.
For the treatment group: C(15, 4) = 15! / (4! * (15 - 4)!) = 1365 ways
For the placebo group: C(10, 2) = 10! / (2! * (10 - 2)!) = 45 ways
The total number of ways to choose six people from 25 is C(25, 6) = 25! / (6! * (25 - 6)!) = 177100
Now, we find the probability by multiplying the combinations for the treatment and placebo groups, and then dividing by the total combinations.
Probability = (C(15, 4) * C(10, 2)) / C(25, 6)
Probability = (1365 * 45) / 177100 = 0.3465
Therefore, the probability that four people received the treatment and two received the placebo is approximately 0.3465 or 34.65%.