The All-Star Pigeons lose $10$ fans every time
they lose a game and gain $100$ fans every time they win. Assume they have
a 30% chance of losing each game and that their performance in each game is
independent. What is the expected value of $X$, the number of fans they will
gain or lose over the next 3 games?
My work:
Let $Y$ represent the number of games the All-Star Pigeons win.
$P(Y=0) = 0.3^3 = .027$
$P(Y=1) = 3 \cdot 0.7 \cdot 0.3^2 = .189$
$P(Y=2) = 3 \cdot 0.7^2 \cdot 0.3 = .441$
$P(Y=3) = .7^3 = .343$
So the expected value for $Y$ is $E(Y) = 0 \cdot .027 + 1 \cdot .189 + 2 \cdot .441 + 3 \cdot .343 = 2.1$
So with this logic the expected number of wins is $2.1$ so I believe that $E(X) = 2.1 \cdot 100 = 210$ fans gained
So the expected value of $X$ is $210$.
My question is, is my methodlogy (and answer) for this question correct? I can't think of any other way to calculate $E(X)$. Could someone here let me know if my thinking is correct? Thank You!
Best Answer
You are nearly there: you need to subtract an extra $0.9\times10=9$ for the games they lose.
Here's another approach, which generalises easily to any number $n$ of games.
Let $X_i$ be the number of fans they gain after game $i$ (which is negative for a loss). Then $$\mathbb E[X_i]=0.7\times100+0.3\times(-10)=67.$$ So by linearity of expectation, $$\mathbb E[X]=\mathbb E[X_1+X_2+X_3]=\mathbb E[X_1]+\mathbb E[X_2]+\mathbb E[X_3]=3\times67=201.$$