In a large crowd,there are three times as many men as women.Three people are chosen at random.Assuming that there are so many people that choosing three has a negligible effect on the proportion of men to women,find the probability that they are (a) all men (b) 2 women and 1 man
p(man) = 3/4
p(woman) = 1/4
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M and N are midpoints of opposite sides of a square ABCD. A point is selected at random in the square.Find the probability that it lies in triangle ADM
To find the probability of each scenario, we first need to determine the total number of men and women in the large crowd.
Let's assume there are M men and W women in the crowd.
Given that there are three times as many men as women, we can write the following equation:
M = 3W
Now, let's determine the total number of people in the crowd:
Total number of people = M + W = 3W + W = 4W
Since we are assuming that choosing three people has a negligible effect on the proportion of men to women, we can treat the selection as a probability problem using combinations.
(a) Probability of selecting all men: To calculate this probability, we need to find the number of ways to choose three men out of the total number of people. Since there are M men and W women, the number of ways would be:
Number of ways to choose all men = C(M,3)
(b) Probability of selecting 2 women and 1 man: Similarly, we need to find the number of ways to choose 2 women out of W and 1 man out of M. The number of ways would be:
Number of ways to choose 2 women and 1 man = C(W,2) * C(M,1)
To find the probabilities, we need to divide the number of ways for each scenario by the total number of possible combinations of choosing any three people from the crowd:
Total number of ways to choose 3 people = C(4W,3)
(a) Probability of selecting all men = C(M,3) / C(4W,3)
(b) Probability of selecting 2 women and 1 man = [C(W,2) * C(M,1)] / C(4W,3)
By plugging in the values for M and W, we can calculate the probabilities.