Chuck-a-luck (also sometimes called birdcage) is a game of chance played with th

Question

Chuck-a-luck (also sometimes called birdcage) is a game of chance played with three 6-sided dice. In the game you can pick any number from 1 to 6 and the three dice are then “rolled” in a cage. If $1 is bet and exactly one of the number that you picked is rolled then you win $1, if exactly two of the number that you picked are rolled you win $2, and if all three of the dice are the number that you picked you win $4 (in each of those cases you also get your initial $1 bet back). If none of your number winds up being rolled you lose your $1 bet. Suppose that you play 5 games of chuck-a-luck and pick the same number each time.

a) What is the probability that triples of your number (all three dice come up your number) occurs at least once?

b) What is your total expected win or loss? Indicate in your answer both the amount and win or loss.

Explanation / Answer

a)the sample space will contain 6^3 = 216 samples

now we have to found the probability of the number that tripple up

probability of getting any tripple number = 1/216 (like 6,6,6 or 1,1,1 which can be one at a time)

probability of non tripple number = 215/216

probability that triple occurs atleast - probability no tripple comes up

n = 5

probability no triple comes up = 5C5*(1/216)^0*(215/216)^5 = 0.977

PROBABILITY OF AT LEAST ONCE TRIPLE = 1 - 0.977 = 0.023

B)E(X) = X1*P(X1)+X2*P(X2)+.....XN*P(XN)

SAMPLE SPACE =

ONE TIME NUMBER = 28*6 = 168

TWO TIME NUMBER = 7*6 = 42

3 TIME NUMBER = 6

PROBABILITY = 1*168/216+2*42/216+4*6/216 = 0.77+0.38+0.11 = 1.26

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