a pool of potential jurors consists of 10 men and 30 women. if three different people are randomly selected from this pool (without replacement), find the probibility that they are all men,
To find the probability that all three people selected are men, we need to determine the number of favorable outcomes (selecting all men) and the total number of possible outcomes.
The number of favorable outcomes is determined by selecting all three men from the pool of 10 men. Since we are selecting without replacement, the number of favorable outcomes can be calculated using the combination formula:
C(n, r) = n! / (r!(n - r)!)
In this case, n = 10 (number of men) and r = 3 (number of men to be selected). Thus, the number of favorable outcomes is:
C(10, 3) = 10! / (3!(10 - 3)!) = 120 / (6 * 5040) = 120 / 720 = 1/6
The total number of possible outcomes is determined by selecting three people from a group of 10 men and 30 women without replacement. Similar to the favorable outcomes, we can determine the number of possible outcomes using the combination formula:
C(n, r) = n! / (r!(n - r)!)
In this case, n = 40 (total number of people) and r = 3 (number of people to be selected). Thus, the number of possible outcomes is:
C(40, 3) = 40! / (3!(40 - 3)!) = 40! / (3! * 37!) = (40 * 39 * 38) / (3 * 2 * 1) = 9880
Therefore, the probability that all three people selected are men can be calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1/6 / 9880
Probability ≈ 0.00010121457489878542
To find the probability that three randomly selected people from the pool are all men, we need to calculate the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
We have a total of 40 people in the jury pool (10 men and 30 women). When we randomly select three people without replacement, we have to consider the order in which they are selected. This can be calculated using combinations. The total number of possible outcomes can be calculated using the combination formula as:
Total number of possible outcomes = C(40, 3) = 40! / (3! * (40-3)!) = 40! / (3! * 37!) = 8,840
Number of favorable outcomes:
We want to select three men from the pool. Since we have 10 men in the pool, we can calculate the number of favorable outcomes using combinations:
Number of favorable outcomes = C(10, 3) = 10! / (3! * (10-3)!) = 10! / (3! * 7!) = 120
Probability:
The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Therefore, the probability of randomly selecting three men from the pool is:
P(All three are men) = Number of favorable outcomes / Total number of possible outcomes
= 120 / 8,840
≈ 0.0136 (rounded to four decimal places) or approximately 1.36%.
The probability of all events occurring is found by multiplying the individual probabilities.
First = 10/40
Second = 9/39
Third = 8/38