Three students from a 22 student class will be selected to attend a meeting. 11 of the students are female.
What is the probability that not all three of the students chosen to attend the meeting are female? =
To find the probability that not all three of the students chosen to attend the meeting are female, we need to find the complement of the event where all three students chosen are female.
First, we need to calculate the total number of ways to choose 3 students out of a class of 22. This is given by the combination formula, denoted as C(n, r), where n is the total number of objects and r is the number of objects to be chosen. In this case, n = 22 and r = 3. So, the total number of ways to choose 3 students can be calculated as:
C(22, 3) = 22! / (3!(22 - 3)!) = 22! / (3!19!) = (22 * 21 * 20) / (3 * 2 * 1) = 22 * 7 * 10 = 1540
Next, we need to calculate the number of ways to choose 3 female students out of the 11 females in the class. This can be calculated as:
C(11, 3) = 11! / (3!(11 - 3)!) = 11! / (3!8!) = (11 * 10 * 9) / (3 * 2 * 1) = 11 * 5 * 3 = 165
Therefore, the probability that all three students chosen are female is:
P(all three students chosen are female) = (Number of ways to choose 3 female students) / (Total number of ways to choose 3 students)
P(all three students chosen are female) = 165 / 1540
To find the probability that not all three students chosen are female, we subtract the probability of all three students being female from 1:
P(not all three students chosen are female) = 1 - P(all three students chosen are female)
P(not all three students chosen are female) = 1 - (165 / 1540)
Finally, we can calculate the probability:
P(not all three students chosen are female) ≈ 0.8935 or 89.35%