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
the tree diagram
pr(man)=1/4
pr(man,man,man)=(1/4)^3
Pr(2w,1m)..
can occur wwm,wmw,mww three ways
pr(2w,1m)=3*(3/4)^2(1/4)
which is the same as 3!/1!2! *(3/4)^2*(1/4)
To solve this problem, we need to determine the total number of men and women in the large crowd, and then calculate the probability of selecting all men or 2 women and 1 man.
Let's assume there are M men and W women in the crowd.
According to the given information, we know that there are three times as many men as women. Therefore, we have the equation: M = 3W.
To find the total number of people in the crowd, we add the number of men and women: Total population = M + W.
Now, let's calculate the probabilities:
(a) Probability of selecting all men:
To calculate this probability, we need to find the number of ways we can select 3 men from the total population of men.
The probability of selecting all men is given by:
P(all men) = [Number of ways to select 3 men] / [Total number of ways to select 3 people]
The number of ways to select 3 men from M men is given by the combination formula: C(M, 3) = M! / (3!(M-3)!).
Similarly, the total number of ways to select 3 people from the total population is given by the combination formula: C(M+W, 3) = (M+W)! / (3!(M+W-3)!).
Therefore, the probability of selecting all men is:
P(all men) = C(M, 3) / C(M+W, 3).
(b) Probability of selecting 2 women and 1 man:
To calculate this probability, we need to find the number of ways we can select 2 women from the total population of women, and 1 man from the total population of men.
The probability of selecting 2 women and 1 man is given by:
P(2 women and 1 man) = [Number of ways to select 2 women] * [Number of ways to select 1 man] / [Total number of ways to select 3 people]
The number of ways to select 2 women from W women is given by the combination formula: C(W, 2) = W! / (2!(W-2)!).
Similarly, the number of ways to select 1 man from M men is given by the combination formula: C(M, 1) = M! / (1!(M-1)!).
Finally, the total number of ways to select 3 people from the total population is given by the combination formula: C(M+W, 3) = (M+W)! / (3!(M+W-3)!).
Therefore, the probability of selecting 2 women and 1 man is:
P(2 women and 1 man) = [C(W, 2) * C(M, 1)] / C(M+W, 3).
Now, using the equation M = 3W, we can substitute the value of M in terms of W to get the final probabilities.
To find the probability, we first need to determine the total number of possible outcomes and the number of favorable outcomes.
Let's denote the number of men as M and the number of women as W. We are given that in a large crowd, there are three times as many men as women, so we can express this as:
M = 3W
The total number of people in the crowd is given by:
Total = M + W = 3W + W = 4W
(a) Probability of selecting all men:
To find the probability of choosing all men, we need to determine the number of favorable outcomes (where all three chosen individuals are men) and divide it by the total number of possible outcomes.
The number of favorable outcomes is equal to the number of ways we can choose three individuals from the group of men (M). This can be computed using the combination formula, also known as "nCr" or "binomial coefficient."
Favorable_outcomes = C(M, 3)
The total number of possible outcomes is equal to the number of ways we can choose three individuals from the entire group of people (Total).
Total_outcomes = C(Total, 3)
Therefore, the probability of selecting all men is:
Probability_all_men = Favorable_outcomes / Total_outcomes = C(M, 3) / C(Total, 3)
(b) Probability of selecting 2 women and 1 man:
To find the probability of selecting two women and one man, we need to determine the number of favorable outcomes (where two chosen individuals are women and one is a man) and divide it by the total number of possible outcomes.
The number of favorable outcomes can be calculated as follows:
Favorable_outcomes = C(W, 2) * C(M, 1)
Similarly, the total number of possible outcomes can be calculated:
Total_outcomes = C(Total, 3)
Therefore, the probability of selecting 2 women and 1 man is:
Probability_2_women_1_man = Favorable_outcomes / Total_outcomes = (C(W, 2) * C(M, 1)) / C(Total, 3)
Please note that to obtain the actual numerical values, you would need to know the specific values of M and W (e.g., if M=30 and W=10, then you can substitute these values into the formulas).