Binomial distribution: what is the probability of getting exactly 3 women from a draw of Y = 1 to 10?

Question

So my question is We choose a certain number Y of different people

1. what is the probability of getting exactly 3 women from a draw of Y = 1 to 10 ?

2. what is the probability of getting at least more women than men from a draw of Y = 1 to 10

Consider the following: there are 45 people, 38 men and 7 women

1. I tried for example when Y = 1; P(3 women) = 1C3 * (7/45)^3 * (38/45)^-2

edit: Y = 3 would work in my eyes, and everything above Y = 3 until Y = 10, but under Y = 3 wouldnt be possible in my opinion, or am i wrong there?

my question now is, would that be correct? I have never calculated something like this with a negative exponent. Consider that 1C3 doesnt work..

2. I tried for example Y = 10; 1 - P( most more women than men) = 1 - 10C3 * (7/45)^7 * (38/45)^3 , but that doesnt look correct in my opinion..

Answer 1

I think certain things should be cleared up before answering.

If the extraction happens with replacement (which I assume), then the binomial distribution expresses the probability of having $k$ successes in $Y$ trials.

In question $1$, if success is the event "extracting a woman" (happening with probability $p$) then the probability of the event in the question is

$${Y\choose{k}}p^k(1-p)^{Y-k} = {10\choose{3}}\left(\frac{7}{45}\right)^3\left(\frac{38}{45}\right)^7$$

Why? Because there are ${10\choose{3}}$ possible ways of extracting exactly 3 women and 7 men out of 10 draw (this is the crucial point, the draws are 10!), and the probability of said event does not change.

As for problem 2, try considering all the possible ways in which the event "more women than men have been extracted" can happen. Note that the events "6 women and 4 men", "7 women and 3 men" ecc... are all mutually disjoint, and so you can apply the law of total probability easily.

Addendum

I strongly suspect that the question being asked in point 1 and 2 is asking what the probability of said events is out of 10 draws. However, let's suppose the question actually refers to all possible number of draws from 1 to 10.

If you look carefully at the formula I have written above, it states the probability of having $k$ successes in $Y$ trials.

Suppose we are dealing with $Y=4$. In what ways can I have more women than men in this number of draws? Well, I could have 0 men and 4 women or 1 man and 3 women. These two events happen with respective probability

$${4\choose{4}}\left(\frac{7}{45}\right)^4\left(\frac{38}{45}\right)^0$$

$${4\choose{3}}\left(\frac{7}{45}\right)^3\left(\frac{38}{45}\right)^1$$

These events are disjoint for any given experiment, so the probability of their union is the sum of their probabilities. Adding them up, we get the desired result.

For each number of draws, you have to make the same considerations. This looks like a rather daunting and tedious exercise, so I would suggest checking if the question was rather not

What is the probability of getting at least more women than men from 10 draws?