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.
Why is it over 4?
all men: Pr(man, man, man)=(3/4)^3
two w, 1 man: ways to get those
WWM
WMW
MWW three ways
pr=3(3/4)(1/4)^2 or 3!/2!(1!) * 3/4*(1/4)^2= same as before, 3(3/4)(1/4)^2
To solve this problem, we need to determine the total number of men and women in the large crowd, and then calculate the probability for each scenario.
Let's assume that there are x women in the crowd. According to the problem, there are three times as many men as women, so the number of men in the crowd would be 3x.
a. Probability of all men:
To calculate the probability of choosing all men, we need to determine how many ways we can choose 3 men out of the total number of men in the crowd (3x).
The total number of ways to select 3 men out of 3x is given by the combination formula: C(3x, 3) = (3x)! / [(3x - 3)! * 3!].
Now, the total number of people in the crowd is 3x (men) + x (women) = 4x.
The probability of choosing all men can be calculated by dividing the number of ways to choose 3 men by the total number of possible combinations of choosing 3 people from the crowd (4x): C(3x, 3) / C(4x, 3).
b. Probability of 2 women and 1 man:
Similarly, to calculate the probability of choosing 2 women and 1 man, we need to determine how many ways we can choose 2 women out of x and 1 man out of 3x.
The number of ways to select 2 women out of x is given by the combination formula: C(x, 2) = x! / [(x - 2)! * 2!], and the number of ways to select 1 man out of 3x is C(3x, 1).
The probability of choosing 2 women and 1 man can be calculated by multiplying the number of ways to choose 2 women and 1 man and dividing it by the total number of possible combinations of choosing 3 people from the crowd: (C(x, 2) * C(3x, 1)) / C(4x, 3).
By plugging in the formulas and simplifying, you can calculate the exact probabilities for each scenario.