A swim team consists of 7 boys and 5 girls. A relay of 4 swimmers is chosen at random from the team members. What is the probability that there are 3 boys on the relay team given that there is 1 girl on the relay team. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.
To find the probability of having 3 boys on the relay team given that there is 1 girl, we need to determine the number of favorable outcomes and the total number of possible outcomes.
First, let's find the total number of possible relay teams. Since a relay team consists of 4 swimmers chosen from the 12 team members, we can calculate this using the combination formula:
Total number of possible relay teams = C(12, 4) = 12! / (4! * (12 - 4)!) = 495
Now, let's determine the number of favorable outcomes where there are 3 boys and 1 girl on the relay team.
The number of ways to choose 3 boys from the 7 available is given by C(7, 3) = 7! / (3! * (7 - 3)!) = 35.
The number of ways to choose 1 girl from the 5 available is given by C(5, 1) = 5! / (1! * (5 - 1)!) = 5.
To find the number of favorable outcomes, we multiply these two values together: 35 * 5 = 175.
Therefore, the probability of having 3 boys on the relay team given that there is 1 girl is:
P(3 boys | 1 girl) = Favorable outcomes / Total outcomes = 175 / 495 ≈ 0.353.
Rounded to the nearest millionth, the probability is approximately 0.353.
To find the probability of having 3 boys on the relay team given that there is 1 girl on the relay team, we first need to determine the total number of possible relays that can be formed.
The total number of team members is 7 boys + 5 girls = 12.
Since the relay team consists of 4 swimmers, we need to calculate "12 choose 4" denoted as 12C4, which represents the number of ways we can choose 4 swimmers from a group of 12.
The formula for "n choose r" is:
nCr = n! / (r! * (n-r)!),
where n! denotes the factorial of n.
Using this formula, we can calculate 12C4 as:
12C4 = 12! / (4! * (12-4)!) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 12,240.
Now, we need to determine the ways we can have 3 boys and 1 girl on the relay team. To calculate this, we multiply the number of ways we can choose 3 boys from a group of 7 (7C3) with the number of ways we can choose 1 girl from a group of 5 (5C1).
7C3 = 7! / (3! * (7-3)!) = (7 * 6 * 5) / (3 * 2 * 1) = 35,
5C1 = 5! / (1! * (5-1)!) = (5 * 4) / (1 * 1) = 20.
To find the total number of ways we can have 3 boys and 1 girl, we multiply 7C3 with 5C1:
Total number of ways = 35 * 20 = 700.
Finally, we can calculate the probability as the number of favorable outcomes divided by the total number of possible outcomes:
Probability = Number of ways to have 3 boys and 1 girl / Total number of possible relays = 700 / 12,240.
Now, we can express the probability either as a fraction or a decimal.