Chuck-a-luck is an old game, played mostly in carnivals and county fairs. To play chuck-a-luck you p

Chuck-a-luck is an old game, played mostly in carnivals and county fairs. To play chuck-a-luck you place a bet, say $1, on one of the numbers 1 through 6. Say that you bet on the number 4. You then roll three dice (presumably honest). If you roll three 4’s, you win $3.00; If you roll just two 4’s, you win $2; if you roll just one 4, you win $1 (and, in all of these cases you get your original $1 back). If you roll no 4’s, you lose your $1. Compute the expected payoff for chuck-a-luck.

Expected payoff for each event = Event Probability * Event Payoff

Expected payoff for 3 matches:
3(1/6 * 1/6 * 1/6) = 3/216 = 1/72

Expected payoff for 2 matches:
2(1/6 * 1/6 * 5/6) = 10/216 = 5/108

Expected payoff for 1 match:
1(1/6 * 5/6 * 5/6) = 25/216

Expected payoff for 0 matches:
-1(5/6 * 5/6 * 5/6) = 125/216

Add all these up:
(3 + 10 + 25 - 125)/216
-87/216 ~ -0.40