You play a gambling game with a friend in which you roll a die. If a 1
or 2
comes up, you win $8.
How much should you lose on any other outcome in order to make this a fair game?
There are six equally likely outcomes when you roll a die. Out of these, two are favorable to you - 1 and 2 - and four are unfavorable - 3, 4, 5, and 6.
Let's call the amount you should lose on any unfavorable outcome "x". For the game to be fair, your expected winnings should be equal to your expected losses.
Expected winnings = Probability of winning x Amount won = (2/6) x $8 = $16/6 = $2.67
Expected losses = Probability of losing x Amount lost = (4/6) x x
For the game to be fair:
Expected winnings = Expected losses
$2.67 = (4/6) x
x = $4
Therefore, for the game to be fair, you should lose $4 on any other outcome apart from 1 and 2.