Cfrgtkky

A group of $n = 10$ men exchange their gloves randomly (altogether there are 20 gloves).
Let $X_i$ be the random variable which attains the value 1 if the $i-th$ man got at least one of his gloves back and it attains the value 0 otherwise. Set $p = E(X_i)$ and $σ^2 = Var(X_i)$.

Observe that $p = frac{37}{190}$ and that $Var (X_i) = frac{5661}{190^2}$.
Let $b$ be the probability that for $ine j$ both the $i-th$ and $j-th$ men got at least one of their gloves back.

Show that $b = frac {1157}{{20 choose 2}{18 choose 2}}$.

1.) Let $X$ be the total number of men who received at least one of their gloves. Find $E(X)$.

2.) Calculate $Var(X).$ You may write it as an expression in $p$, $σ^2$ and $b.$

At first I just thought that $E(X)=np=10frac{37}{190}$ and then $Var(X)=npq=10cdot 37cdotfrac{153}{190^2}$ but now I am not sure about that, as the events do intersect: if someone takes gloves of someone else, then someone else has nothing to choose from. However, I do not know how to count these. Could you help me, please? Thanks

Your calculations of $p$, $sigma^2$ and $mathbb EX$ are okay.

Covariance is bilinear, so that by symmetry:

$$mathsf{Var}(X)=mathsf{Cov}(sum_{i=1}^{10}X_i,sum_{j=1}^{10}X_j)=sum_{i=1}^{10}sum_{j=1}^{10}mathsf{Cov}(X_i,X_j)=10mathsf{Var}(X_1)+90mathsf{Cov}(X_1,X_2)=$$$$10sigma^2+90(mathbb EX_1X_2-mathbb EX_1mathbb EX_2)=10sigma^{10}+90(P(X_1=X_2=1)-P(X_1=1)P(X_2=1))=$$$$10sigma^2+90(b-p^2)$$

Here $b=P(X_1=X_2=1)$ so it remains to find $P(X_1=X_2=1)$

edit:

Let $Z_{1}$ denote the number of gloves taken by person 1 that
belong to person 1.

Let $Y_{1}$ denote the number of gloves taken by person 1 that belong
to person 2.

Then:

$$Pleft(X_{1}=X_{2}=1right)=$$$$Pleft(X_{2}=1mid Z_{1}=2right)Pleft(Z_{1}=2right)+$$$$Pleft(X_{2}=1mid Z_{1}=1wedge Y_{1}=0right)Pleft(Z_{1}=1wedge Y_{1}=0right)+$$$$Pleft(X_{2}=1mid Z_{1}=1wedge Y_{1}=1right)Pleft(Z_{1}=1wedge Y_{1}=1right)$$

This with

• $Pleft(X_{2}=1mid Z_{1}=2right)=1-Pleft(X_{2}=0mid Z_{1}=2right)=1-frac{16}{18}frac{15}{17}$




• $Pleft(X_{2}=1mid Z_{1}=1wedge Y_{1}=0right)=1-Pleft(X_{2}=0mid Z_{1}=1wedge Y_{1}=0right)=1-frac{16}{18}frac{15}{17}$




• $Pleft(X_{2}=1mid Z_{1}=1wedge Y_{1}=1right)=1-Pleft(X_{2}=0mid Z_{1}=1wedge Y_{1}=1right)=1-frac{17}{18}frac{16}{17}$




• $Pleft(Z_{1}=2right)=frac{2}{20}frac{1}{19}$




• $Pleft(Z_{1}=1wedge Y_{1}=0right)=frac{2}{20}frac{16}{19}+frac{16}{20}frac{2}{19}$




• $Pleft(Z_{1}=1wedge Y_{1}=1right)=frac{2}{20}frac{2}{19}+frac{2}{20}frac{2}{19}$