Homework 3: Analyzing the Casino Game of Craps

Due Monday April 18 (send by email to pasquale at cs.ucsd.edu)


1. The house edge for a pass line bet in Craps is 1.41%.  Derive this value and show your work.  Here are some hints to guide you:

a. On the opening roll, a 7 or 11 wins: what is the probability of a win on the first roll?  Call this probability A.

b. On the opening roll, a 2, 3, or 12 loses: what is the probability of a loss on the first roll?  Call this probability B.

c. What is the probability of a 4 or 10 "point" being made on the first roll?  Call this probability C.

d. What is the probability of a 5 or 9 "point" being made on the first roll?  Call this probability D.

e. What is the probability of a 6 or 8 "point" being made on the first roll?  Call this probability E.

We now have probabilities for all possible first rolls.  As a check, they should all add up to 1.  If they don't, you made some error above.

f. Assuming a 4 or 10 "point" was made on the first roll, what is the probability that this point will come out BEFORE a 7?  Call this probability F.

g. Assuming a 5 or 9 "point" was made on the first roll, what is the probability that this point will come out BEFORE a 7?  Call this probability G.

h. Assuming a 6 or 8 "point" was made on the first roll, what is the probability that this point will come out BEFORE a 7?  Call this probability H.

You can now compute the probability P of a win: A + C*F + D*G + E*H.

i. What are the fair odds for a pass line bet?

j. What is the house edge (given that the casino pays 1 to 1 for a pass line bet)?

2. Read the FAQ on Craps located at http://www.faqs.org/faqs/gambling-faq/craps/.  Learn how to place all the bets below so that you are ready to play at our next meeting.  For each of the bets below, what are the fair odds (you need to compute these), casino odds (look them up in the FAQ) and house edges (look them up in the FAQ, and if you have the time, compute and check that they match what are in the FAQ):

a. "Odds" on 4 or 10 point:

b. "Odds" on 5 or 9 point:

c. "Odds" on 6 or 8 point:

d. "Place" on 4 or 10 point:

e. "Place" on 5 or 9 point:

f. "Place" on 6 or 8 point:

g. "Field" (with 2 and 12 paying 2 to 1):

h. "Big 6 or 8":

i. "Proposition" on 2, 3, or 12:

j. "Proposition" on 7:

k. "Proposition" on 11:

l. "Hard" 4:

m. "Hard" 6:

n. "Hard" 8:

o. "Hard" 10:

p. "Any craps":



THE FOLLOWING ARE ADVANCED EXERCISES AND ARE NOT REQUIRED.  DO THEM IF YOU HAVE TIME, AS THEY WILL SHARPEN YOUR ABILITY TO COMPUTE PROBABILITIES AND HOUSE EDGES.

3. What is the house edge against the gambler who always takes 2x odds (i.e., if a point is made, you bet twice the amount that was placed on the pass line)?

4. Rather than making a pass line bet, you bet on "don't pass", which means you win whenever a pass line bet loses, and vice-versa, except for a 12 on the opening roll, which is a "push", i.e., you neither win nor lose, you simply get back what you bet and no more.  What are the fair odds, and what is the house edge (compare this to a pass line bet).