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

number of women ---- x

number of men ------ 3x
prob(choosing one man) =3x/4x = 3/4
prob(choosing one woman) = x/4x = 1/4

a) prob(3 men) = (3/4)^3 , you said that a choice has no effect on the next choice
= 27/64

b) 2 women, 1 man ---> could be WWM, WMW, or MWW
each one has a prob. of (1/4)(1/4)(3/4) = 3/64

so prob(2W,1M) = 3*(3/64) = 9/64

Thanks

Thank you very much

To find the probability, we need to determine the total number of possible outcomes and the number of favorable outcomes for each case.

Let's start with finding the total number of possible outcomes. Since three people are chosen at random, and the proportion of men to women remains negligible, we can assume that the ratio will be the same for the entire crowd.

Let's assume there are a total of 'x' women in the crowd. Then, the number of men in the crowd will be three times the number of women, which is 3x.

The total number of people in the crowd will be the sum of men and women: x + 3x = 4x.

The total number of possible outcomes is given by selecting three people from a group of 4x. We can calculate this using a combination formula:

Total number of possible outcomes = C(4x, 3) = (4x)! / (3!(4x - 3)!)

Now let's calculate the favorable outcomes for each case:

a) All men:
Since there are 3 times as many men as women, the probability of selecting a man is 3x / 4x = 3/4.
To select three men, we need to multiply the probabilities:
Probability of selecting three men = (3/4) * (3/4) * (3/4) = 27/64.

b) 2 women and 1 man:
The probability of selecting a woman is x / 4x = 1/4.
The probability of selecting a man is 3x / 4x = 3/4.
To select 2 women and 1 man, we need to multiply the probabilities:
Probability of selecting 2 women and 1 man = (1/4) * (1/4) * (3/4) + (1/4) * (3/4) * (1/4) + (3/4) * (1/4) * (1/4) = 9/64 + 9/64 + 9/64 = 27/64.

Therefore, the probability of selecting three men is 27/64, and the probability of selecting 2 women and 1 man is also 27/64.