Question

A certain game consists of rolling a single fair die and pays off as follows: $6 for a 6, $4 for a 5, $1 for a 4, and no payoff otherwise. Find the expected winnings for this game.

Answer 1

The expected winnings are given by the following formula:

$E(x)=\sum ^{}_{}x\cdot P(x)$

were x all the gains and P(x) their individual probabilities.

Then, the probability of getting a 6 is 1/6, the probability of getting a 5 is 1/6 and the probability of getting a 4 is 1/6, then the expected winnings for this game are:

$\begin{gathered} E(x)=6\cdot\frac{1}{6}+4\cdot\frac{1}{6}+1\cdot\frac{1}{6} \\ E(x)=\frac{6}{6}+\frac{4}{6}+\frac{1}{6} \\ E(x)=\frac{6+4+1}{6} \\ E(x)=\frac{11}{6} \\ E(x)=1.83 \end{gathered}$

The expected winnings are $1.83.