Chuck-A-Luck is a carny game in which 3 dice are tossed. You pick a number from 1 to 6. If one of the dice shows your number you gain the value of your bet; if two of the dice show your number you gain twice the value of your bet and if three of your number show, you gain three times the value of your bet. Otherwise you lose the amount you bet.
What is your expected loss for each game?
(In reply to
Puzzle Answer by K Sengupta)
If the chosen number shows up a single time, then the gain
= 3*(1/6)*(5/6) ^2 = 75/216
If it shows up twice, then the gain
= 2*(1/6)^2*(5/6)*3 = 30/216
If it shows up thrice, then the gain
= 3*(1/6)^3 = 3/216
So, the total gain if the number shows up once, twice or thrice
=(75+30+3)/216 = 108/216
If the chosen number does not show up, then the loss
=(5/6)^3 = 125/216
Consequently, the required expected loss for each game
= (108-125)/216
= -17/216
Edited on April 29, 2022, 9:23 pm
Explanation to the Puzzle Answer
