Craps is one of the most popular games in casinos today and has roots dating all
ID: 3153556 • Letter: C
Question
Craps is one of the most popular games in casinos today and has roots dating all the way back to the Crusades. At its core, someone playing craps is simply betting on a roll of 2 dice. You may assume the rolls are independent and the dice are fair.
(a) What is the sample space in this example? Construct a 6-by-6 table to fully describe the sample space.
(b) Assuming fair dice, what is the probability distribution for the experiment? The first phase of a round of craps is known as the come-out. The player rolling the dice (shooter) places a bet on the pass line. This bet has three outcomes. The shooter loses if the sum of the two dice is 2, 3, or 12 (called craps). The shooter wins if the sumis7or11arerolled. Ifa4,5,6,8,9,or10arerolledthenthegamemovestothe point phase.
(c) What is the probability of winning during the come-out phase?
(d) What is the probability of losing during the come-out phase?
(e) What is the probability of moving to the point phase?
If the round of craps proceeds into the point phase, then the number that was rolled during the come-out is aptly called the point. The goal of the most basic bet in craps (the pass line bet), is to roll the point before a 7 is rolled. If the point is rolled, then the player wins (amount varies based on the value of the point). If a seven is rolled, every bet on the table is cleared with almost all being losers (there are special bets that pay on this situation but we wont look at those here).
(f) If the casino only wins when one number is rolled, why do they still make money?
(g) What would the most advantageous values for the point be for a player?
Explanation / Answer
2 dice are rolled. the rolls are independent.
each die has 6 faces 1,2,3,4,5,6
a) hence the sample space here is the all possible outcomes of the two dice.
hence the sample space is
where the first number denotes the outcome of the first die and the second number denotes the same for the second die.
b) assuming fair die then we have the probability of each outcome of each die is 1/6
and since the rolls are independent hence the probability of each outcome of each sample point of the sample space is (1/6)*(1/6)=1/36
hence the probability distribution of the experiment is discrete uniform. with the constant probability 1/36
the shooter loses if the sum of two dice are 2,3 or 12
now from the sample space it is clear that
the sum will be 2 if (1,1) occurs
the sum will be 3 if (1,2) or (2,1) occurs
sum will be 12 if (6,6) occurs
so if either of (1,1),(2,1),(1,2) or (6,6) occurs then the shooter loses
hence the probability of losing during the come-out phase is
P[losing during the come out phase]=P[(1,1)]+P[(2,1)]+P[(1,2)]+P[(6,6)]=1/36+1/36+1/36+1/36=4/36=1/9
[answer (d)]
the shooter wins if 7 or 11 occurs as the sum
now the sum will be 7 if either of (6,1),(1,6),(5,2),(2,5),(3,4),(4,3) occurs
sum will be 11 if either of (5,6) or (6,5) occurs
so the probability of winning during the come-out phase is
P[winning during the come-out phase]=P[(6,1)]+P[(1,6)]+P[(5,2)]+P[(2,5)]+P[(3,4)]+P[(4,3)]+P[(5,6)]+P[(6,5)]
=8*(1/36)=2/9 [answer (c)]
now if the sum is 4,5,6,8,9 or 10 the game moves to the point phase.
now the sum can either be 2,3,4,5,6,7,8,9,10,11 or 12
out of which 2,3,12 denoting losing 7,11 denotes winning and the rest denotes moving to the point phase.
since total probability is 1. hence the probability of moving to the point phase can be determined by subtracting the probabilities of winning and losing from total probability
hence P[moving to the point phase]=1-P[losing during the come out phase]-P[winning during the come-out phase]
=1-1/9-2/9=1-3/9=1-1/3=2/3 [answer]
1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6