Expected Value and Variance of Groups with All Women

Question

The scenario given by the problem is as follows:

Suppose a class has seventy-five students, with twenty-five men and fifty women. All the students have been randomly assigned into twenty-five study groups of three students each.

Q1: Find the probabilities for the events 1 = “the first group has three women”, and 12 = “the first group has three women and the second group has three women”.

My ans:

The probability for $W1 = {50\choose 3} {25\choose 0} / {75\choose3} = 0.290$

The probability for $W2 = ({50\choose 3} {25\choose 0} / {75\choose3}) * ({47\choose3} {25\choose0} / {72\choose3}) = 0.0789$

Q2: Consider the number of groups that have three women, . Find its expected value and variance.

My ans:

I am unsure as to whether n should be 16 (to denote the number of groups of 3 women that can be formed), or 25 (the total number of groups formed). My preliminary thought is n should be 25 since we defined A1,...,An to denote the event that the nth group has 3 women. However, since at most 16 groups of all women can be formed, I am unsure how to compute the probability of A17 to A25 if n=25.

I am also unsure as to the probability of Ai because the 16 groups with 3 women can be arranged in any order within the 25 groups.

Regarding the variance, I believe the following formula should give the variance:

However, I am unsure as to which group Aj pertains to and the limits of the second summation.

Would really appreciate it if someone can provide an answer to the question and how to evaluate the variance. Thank you for your help in advance.

Answer 1

I am unsure as to whether n should be 16 (to denote the number of groups of 3 women that can be formed), or 25 (the total number of groups formed). My preliminary thought is n should be 25 since we defined A1,...,An to denote the event that the nth group has 3 women.

That is the correct thought.

However, since at most 16 groups of all women can be formed, I am unsure how to compute the probability of A17 to A25 if n=25.

It does not matter. The marginal probability that any particular group has three women is identical for all groups.

I am also unsure as to the probability of Ai because the 16 groups with 3 women can be arranged in any order within the 25 groups.

There are many ways the 50 women may be spread among the study groups. But all you need to calculate is the probability that this (points randomly) group will have three women.

The probability is for obtaining three from fifty women when selecting three from the seventy-five students.

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The rest is the Linearity of Expectation, and the fact that it works whether the sampling is independent or dependent . That is why it is so useful.

$$\begin{align}\mathsf E(\sum_{k=1}^{25} X_k)&=\sum_{k=1}^{25} \mathsf E(X_k)\\[1ex]&=25\times\mathsf P(A_1)\end{align}$$

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Similarly, Variance is evaluated through the Bilinearity of Covariance, though it is slightly more complicated.

$$\begin{align}\mathsf {Var}(\sum_{k=1}^{25} X_k)&=\mathsf{Cov}(\sum_{i=1}^{25} X_i, \sum_{j=1}^{25} X_j)\\[1ex]&=\sum_{i=1}^{25}\sum_{j=1}^{25}\mathsf{Cov}(X_i, X_j)\\&~~\vdots\end{align}$$

Here the dependency matters, as the covariance of distinct groups is not zero (and also different from the variance of a single group).$$\mathsf {Cov}(X_i,X_j)=\begin{cases}\mathsf P(A_i)-\mathsf P(A_i)^2&:& i\in[[1;25]], i=j\\[1ex]\mathsf P(A_1,A_j)-\mathsf P(A_1)\,\mathsf P(A_j)&:&\langle i,j\rangle\in[[1;25]]^2, i\neq j\\[1ex]0&:&\text{elsewhere}\end{cases}$$