Question

You play a game where you roll a single die. You pay $1 to play, and the payouts are $0.50 if you roll an

even number, $2 if you roll a 1, and $1 if you roll a 3 or 5. What are the odds for winning money if you play this game? Show your work and Explain.

Answer 1

Let W be the random variable representing your winnings from playing the game. Then

`P(W=w)=\begin{cases}\text{prob. of rolling an even number}=\frac12&\text{if }w=\$0.50-\$1=-\$0.50\\\text{prob. of rolling a 1}=\frac16&\text{if }w=\$2-\$1=\$1\\\text{prob. of rolling 3 or 5}=\frac13&\text{if }w=\$1-\$1=\$0\\0&\text{otherwise}\end{cases}`

In short, you have a 1/6 chance of profiting from the game, and a 5/6 chance of losing money. So the odds of winning are (1/6)/(5/6) = 1/5 or 1 to 5.