# A question about different pairs that are formed from a set of 16 different poeple such that...

I got the following problem:

Given a set of 16 different people.

We partition the people into pairs of two.

Each pair needs to accomplish a task.

And the probability that a pair accomplishes the task is $0.8$.

(1) Find the variance of the discrete random variable $X$ that denotes the number of pairs that accomplished the task.

(2) Find the variance of the discrete random variable $Y$ that denotes the number of people that accomplished the task.

(3) If before the start of the mission, each pair is given a $100\$$and if the pair accomplishes the task than the pair get an additional 100\$$, if the pair fails to accomplish the task than the pair pays$50\$$from the amount given (the 100\$$ given at the start of the task).

Find the variance of the discrete random variable $Z$ that denotes the total amount of money the 8 pairs hold at the end of the task.

Since each pair has probability of $0.8$ to accomplish the task and since the are $8$ pairs, we get that $X$ is a binomial random variable with parameters $(8,0.8)$ (I.e. $X$~$B(8,0.8)$) and so the variance of $X$ is:

$Var(X)= 8\times 0.8\times (1-0.8)=1.28$

(Please correct me if I wrong)

But in the case of (2) and (3), I got stuck and I don't know how to proceed.

Thanks on any help.

• Hint for (b): $Var(aX+b)=a^2Var(X)$ whare $a$ and $b$ are constants Commented Dec 11, 2014 at 8:40
• Your answer on (1) is correct. The hint of @Henry can also be used for (3). Just write $Y$ and $Z$ as linear forms $aX+b$. Commented Dec 11, 2014 at 9:08
• Thanks a lot. I solved it. Commented Dec 11, 2014 at 10:51

Notice that $Y=2X$ and so $Var(Y)=Var(2X)=2^2Var(X)=4\times 1.28=5.12$
Notice that $Z=100\times 8 + 100\times X -50\times(8-X) = 150X+400$ and so $Var(Z)=Var(150X+400)=150^2Var(X)= 150^2\times 1.28=28800$