Chuck-a-luck
You bet 1 dollar on a number from 1 to 6, and three fair dice are rolled. If your number shows on or of the dice, you get your dollar back plus dollars. If it shows on none, you lose your dollar.
Find your expected net gain per game.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Find the probability that your number shows on exactly 0, 1, 2 or 3 dice.
There are equally likely outcomes. In how many does your number never show?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Count the outcomes by the number of dice showing your number. The game looks fair, since the expected number of matching dice is , but you only collect on the matches while you lose the whole stake when there is none.
Step 1: the counts. Out of 216 outcomes, your number shows on
- no die: ;
- exactly one die: ;
- exactly two dice: ;
- all three: .
Step 2: the expectation. That is about . You lose about 7.9 cents per dollar.
Why it looks fair. The intuition “three dice, each matches with probability , so I win half a dollar for every half dollar I risk” double counts: when two or three dice match, you win several dollars but only once avoid losing your stake. Matches bunch together, and the chance of no match at all is more than . The game becomes fair if two matches pay 3 to 1 and three matches pay 5 to 1.