Maths

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
total = 4x
so prob(choosing a woman) = x/4x
prob(choosing a man) = 3x/4x

prob(choosing 3 men of 3) = C(3,3)(3x/4x)^3 (x/4x)^0 = 1(27/64)(1) = 27/64
or
prob(choosing 3 men) = (3x/4x)( (3x-1)/(4x-1) )( (3x-2)/(4x-2) ) = 3x(3x-1)(3x-2) / (4x(4x-1)(4x-2) )
= (27x^3 - 27x^2 + 6x)/(64x^3 - 48x^2 + 8x)

now we want the
lim [ (27x^3 - 27x^2 + 6x)/(64x^3 - 48x^2 + 8x) ] as x ---->∞
= 27/64

2 women, 1 man:
= C(3,2) (1/4)^2 (3/4)^1 = 3(1/16)(3/4) = 9/64

or, as before
prob = (x/4x)*(x-1)/(4x-1)*(3x)/4x-2) + (3x)/4x-2)*(x/4x)*(x-1)/(4x-1) + (x/4x)*(3x)/4x-2)*(x-1)/(4x-1)
prob = Lim [3(x)(x-1)(3x)/(4x(4x-1)(4x-2) ) as x ---->∞
= lim (9x^3 - 9x^2)/(64x^3 - 48x^2 + 8x) as x ---->∞
= 9/64

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

Let's first find the total number of possible outcomes:
Since we are choosing three people at random, regardless of gender, we can choose from the entire crowd. Therefore, the total number of possible outcomes is the total number of people in the crowd choose 3.

Now, let's find the number of favorable outcomes for each case:

(a) All Men:
Since there are three times as many men as women, let's assume there are m men and w women in the crowd. Therefore, we have m = 3w.
To calculate the number of favorable outcomes for this case, we need to select 3 men from the m men in the crowd. This can be done by calculating m choose 3.
The number of favorable outcomes for case (a) is given by: mC3

(b) 2 Women and 1 Man:
To calculate the number of favorable outcomes for this case, we need to select 2 women from the w women in the crowd and select 1 man from the m men in the crowd. This can be done by calculating w choose 2 multiplied by m choose 1.
The number of favorable outcomes for case (b) is given by: wC2 * mC1

Now that we have the total number of possible outcomes and the number of favorable outcomes for each case, we can calculate the probabilities:

(a) P(all men) = Number of favorable outcomes for all men / Total number of possible outcomes
= mC3 / (m + w)C3

(b) P(2 women and 1 man) = Number of favorable outcomes for 2 women and 1 man / Total number of possible outcomes
= wC2 * mC1 / (m + w)C3

Note that we need to substitute the value of m = 3w in both these probabilities.

By using these formulas, you can calculate the probabilities (a) and (b) given the number of men and women in the crowd.