A football team consists of 20 each freshmen and sophomores, 15 juniors and 10 seniors. Four players are selected at random to serve as captains. Find the probability that:
a) all four are seniors
b) There is one each: freshman, sophomore, junior and senior.
c) There are 2 sophomores and 2 freshmen
d) At least one of the students is a senior
Total players = 65
a) 10/65, 9/64....
b) 20/65, 20/64, 15/63, 10/62 (multiply)
Use the same principles as previous post.
To solve this problem, we need to first find the total number of possible outcomes, which is the number of ways to choose 4 players from a total of 55 players.
Total number of possible outcomes = Combination(55, 4)
a) To find the probability that all four captains are seniors, we need to find the number of ways to choose 4 seniors from a total of 10 seniors.
Number of ways to choose 4 seniors = Combination(10, 4)
Probability (all four captains are seniors) = Number of ways to choose 4 seniors / Total number of possible outcomes
b) To find the probability of selecting one freshman, one sophomore, one junior, and one senior as captains, we need to find the number of ways to choose 1 student from each grade level.
Number of ways to choose 1 freshman = Combination(20, 1)
Number of ways to choose 1 sophomore = Combination(20, 1)
Number of ways to choose 1 junior = Combination(15, 1)
Number of ways to choose 1 senior = Combination(10, 1)
Probability (one each: freshman, sophomore, junior, senior) = (Number of ways to choose 1 freshman * Number of ways to choose 1 sophomore * Number of ways to choose 1 junior * Number of ways to choose 1 senior)/ Total number of possible outcomes
c) To find the probability of selecting 2 sophomores and 2 freshmen as captains, we need to find the number of ways to choose 2 students from each grade level.
Number of ways to choose 2 sophomores = Combination(20, 2)
Number of ways to choose 2 freshmen = Combination(20, 2)
Probability (2 sophomores and 2 freshmen) = (Number of ways to choose 2 sophomores * Number of ways to choose 2 freshmen)/ Total number of possible outcomes
d) To find the probability of at least one senior being chosen as captain, we need to find the number of ways to choose 1, 2, 3, or 4 seniors from a total of 10 seniors.
Number of ways to choose 1 senior = Combination(10, 1)
Number of ways to choose 2 seniors = Combination(10, 2)
Number of ways to choose 3 seniors = Combination(10, 3)
Number of ways to choose 4 seniors = Combination(10, 4)
Probability (at least one senior being chosen) = (Number of ways to choose 1 senior + Number of ways to choose 2 seniors + Number of ways to choose 3 seniors + Number of ways to choose 4 seniors) / Total number of possible outcomes
To solve these probability problems, we'll need to first determine the total possible outcomes and then find the favorable outcomes for each scenario.
a) Probability that all four are seniors:
The total number of players in the football team is 20 freshmen + 20 sophomores + 15 juniors + 10 seniors = 65 players.
The number of ways to choose 4 seniors from the 10 available is given by the combination formula: C(10, 4) = 10! / (4! * (10-4)!) = 210.
The total number of ways to choose any 4 players from the 65 available is given by C(65, 4) = 65! / (4! * (65-4)!) = 65 * 64 * 63 * 62 / (4 * 3 * 2 * 1) = 7,365.
So, the probability that all four of the selected captains are seniors is 210/7,365 ≈ 0.0285 or 2.85%.
b) Probability that there is one each: freshman, sophomore, junior, and senior:
For this scenario, we need to choose one player from each grade level.
The number of ways to choose one freshman from the 20 available is C(20, 1) = 20.
Similarly, the number of ways to choose one sophomore from the 20 available is C(20, 1) = 20.
The number of ways to choose one junior from the 15 available is C(15, 1) = 15.
The number of ways to choose one senior from the 10 available is C(10, 1) = 10.
To calculate the total number of outcomes for this scenario, we multiply the number of choices from each category: 20 * 20 * 15 * 10 = 60,000.
So, the probability of having one freshman, one sophomore, one junior, and one senior as captains is 60,000/7,365 ≈ 0.8143 or 81.43%.
c) Probability that there are 2 sophomores and 2 freshmen:
For this scenario, we need to choose 2 players from the sophomore group of 20 and 2 players from the freshman group of 20.
The number of ways to choose 2 sophomores from the 20 available is C(20, 2) = 190.
Similarly, the number of ways to choose 2 freshmen from the 20 available is C(20, 2) = 190.
The total number of outcomes for this scenario is given by C(65, 4) = 7,365.
So, the probability of having 2 sophomores and 2 freshmen as captains is (190 * 190) / 7,365 ≈ 0.3208 or 32.08%.
d) Probability that at least one of the players is a senior:
To calculate the probability that at least one player is a senior, we'll find the complement probability (i.e., the probability that none of the players are seniors) and subtract it from 1.
The number of ways to choose 4 non-seniors is given by C(55, 4) = 5,435.
The total number of outcomes for this scenario is given by C(65, 4) = 7,365.
So, the probability that at least one player is a senior is 1 - (5,435 / 7,365) ≈ 0.2615 or 26.15%.
Remember, the probability of an event is a number between 0 and 1, where 0 indicates impossible and 1 indicates certain.