Craps

This discussion will looking at the following topics of the game of Craps:

Conclusions

Understanding the Dice

Craps is a game where two dice are thrown and the player has many choices for the bet.  First note that there are 36 different possibilities for the outcome of the two dice, as shown in the table below

 1 2 3 4 5 6 1 (1,1) = 2 (1,2) = 3 (1,3) = 4 (1,4) = 5 (1,5) = 6 (1,6) = 7 2 (2,1) = 3 (2,2) = 4 (2,3) = 5 (2,4) = 6 (2,5) = 7 (2,6) = 8 3 (3,1) = 4 (3,2) = 5 (3,3) = 6 (3,4) = 7 (3,5) = 8 (3,6) = 9 4 (4,1) = 5 (4,2) = 6 (4,3) = 7 (4,4) = 8 (4,5) = 9 (4,6) = 10 5 (5,1) = 6 (5,2) = 7 (5,3) = 8 (5,4) = 9 (5,5) = 10 (5,6) = 11 6 (6,1) = 7 (6,2) = 8 (6,3) = 9 (6,4) = 10 (6,5) = 11 (6,6) = 12

Notice that the sums can range from 2 through 12.  Below are the frequency counts of each sum:

 Sum 2 3 4 5 6 7 8 9 10 11 12 Frequency 1 2 3 4 5 6 5 4 3 2 1

Notice that the probability of rolling a seven is 6/36 = 1/6.  This Table is also shown as a frame to the right since it will be used in most of the calculations.

The Field Bet

The Field Bet is a one roll bet that wins or loses on the one roll.  If you roll a 3,4,9,10, or 11 you win even money.  If a 2 rolls you are paid double.  If a 12 rolls, you are paid triple.  If 5,6,7,or 8 rolls, you lose.  Assume you have a \$100 Field Bet.  Let's compute the expected value.  Below is the table that shows the payouts and their respective probabilities:

 Winnings 100 200 300 -100 Probability 14/36 1/36 1/36 20/36

For example, there are 14 rolls of 3,4,9,10,11 out of 36 total possible rolls.  Hence the probability of even money (100) is 14/36

Hence the expected value is:

100(14/36) + 200(1/36) + 300(1/36) + (-100)(20/36)  =  -2.78

Hence you can expect on the average to lose \$2.78 per wager.

The Pass Line

When you bet on the Pass Line, you win even money if the first roll is a 7 or 11.  You lose you money if the first roll is a 2, 3, or 12.  Notice that the probability that you will win on the first roll is

P(Win on first roll) = 8/36  =  0.222222

What if the first roll is not a 2,3,7,11, or 12?  For example, if the first roll is an 8 (called your point), then you win if an 8 comes up before a 7 on any of the future rolls.  To compute the probability of winning given that the shooter rolled an 8 on the first roll, notice that there are 11 outcomes that make you win or lose on the next roll.  Of these outcomes, 5 make you win and 6 make you lose.  Hence this probability is 5/11.  Since the probability of getting an 8 on the first roll is 5/36, we have

P(Win via a point of 8)  =  (5/36)(5/11)  =  0.063131

Similarly, we can find the probabilities of winning with other points:  4,5,6,9,10:

P(Win via a point of 4)  =  (3/36)(3/9)  =  0.027778

P(Win via a point of 5)  =  (4/36)(4/10)  =  0.044444

P(Win via a point of 6)  =  (5/36)(5/11)  =  0.063131

P(Win via a point of 9)  =  (4/36)(4/10)  =  0.044444

P(Win via a point of 10)  =  (3/36)(3/9)  =  0.027778

To find the probability of winning, just add up all these probabilities:

P(Win)  =  0.222222 + 0.027778 + 0.044444 + 0 .063131 + 0.063131 + 0.044444 + 0.027778

=  0.493

We can calculate the probability of losing by subtracting from 1:

P(Lose)  =   1 - 0.493  =  0.507

Suppose that you place \$100 on the Pass Line, then the expected value is

(100)(0.493) + (-100)(0.507)  =  -1.40

Because of round off error, the actual expected value is -1.41.

That is you can expect to lose on average about \$1.41 per bet.  Notice that the Field Bet is much worse than the Pass Line bet.

The Come Line

The Come Line bet is the same as the pass line except that the bet is placed at any time.  Therefore all the calculations for the Pass Line bet apply to the Come Line bet.

Don't Pass

The Don't Pass bet is almost the opposite of the Pass Line bet.  You are the House with the exception that if a 12 appears on the first roll you push instead of win.  Since we have done most of the calculations already, we will use these.  Notice that the probability of getting a 12 on the first roll is

P(12 on the first roll)  =  1/36  =  0.02778

To find the probability of winning the Don't Pass bet, we subtract 0.028 from the probability of losing on the Pass Line bet (0.507).

P(Winning)  =  0.507 - 0.028  =  0.479

Now let's compute the expected value for a \$100 wager:

(100)(0.479) + (-100)(0.493) + (0)(0.02778)  =  -1.40

You can expect to lose on average \$1.40 per bet.

Note:  the Don't Pass bet is unpopular since you are rooting for everyone else to lose.  I would not recommend it if you want to make friends while gambling.

The Don't Come

The Don't Come bet is the same as the Don't Pass except that the bet is placed at any time.  Therefore all the calculations for the Don't Pass bet apply to the Don't Come bet.

Taking Odds

Taking odds involves placing additional money after the first roll came up a 4,5,6,8,9, or 10 that will give you "true odds" for this number.  For example if the point is a 5, then the odds are 3 to 2, since there are six 7's and four 5's.  Note that 6/4 = 3/2.  Since the payback that you receive matches the true odds, this is an even bet.  If you enjoy gambling, then this is the only "free" bet available.  That is over the long run, the Taking Odds bet will not result in losing or winning any money.

Laying Odds

The Laying Odds bet bet is the reverse of the Taking Odds bet and is allowed when you have a Don't Pass or Don't Come bet.  Similarly, the Laying Odds bet is a "free" bet, that is, over the long run the Laying Odds bet will not result in losing or winning any money.

Big 6 or Big 8

The Big 6 bet is where you bet that a 6 will occur before a 7.  It pays even money.  To compute the probability of winning this bet, notice that there are five 6's and six 7's.  Hence

P(Winning a Big 6)  =  5/11  =  .45454

The probability of losing is

P(Losing a Big 6)  =  1 - .4545  =  .54546

If you wager \$100, then the expected value is

(100)(.45454) + (-100)(.54546)  =  -9.09

So that you can expect to lose on the average \$9.09

The Big 8 bet has the same probability and expected value, since the there are the same number of 8's as 6's.

Proposition Bet

A proposition bet involves betting that you choice will occur on the next roll.  Below is a discussion of each of  the proposition bets:

Any Craps:

You bet that the next roll will be either a 2, 3, or 12.  The payout is 7 to 1.  Compute

P(Winning Any Craps)  =  4/36  =  1/9

P(Losing Any Craps)  =  1 - 1/9  =  8/9

The expected value for a \$100 wager is

(700)(1/9) + (-100)(8/9)  =  -1/9  =  -11.11

Hence on a \$100 bet, you will lose on the average \$11.11.

Any Seven:

You bet that the next roll will be a 7.  The payout is 4 to 1.  Compute

P(Winning Any Seven)  =  6/36  =  1/6

P(Losing Any Seven)  =  1 - 1/6  =  5/6

The expected value for a \$100 wager is

(400)(1/6) + (-100)(5/6)  =  -16.67

Hence on a \$100 bet, you will lose on the average \$16.67.

Eleven:

You bet that the next roll will be an 11.  The payout is 15 to 1.  Compute

P(Winning Eleven)  =  2/36  =  1/18

P(Losing Eleven)  =  1 - 1/18  =  17/18

The expected value for a \$100 wager is

(1500)(1/18) + (-100)(17/18)  =  -11.11

Hence on a \$100 bet, you will lose on the average \$11.11.

Ace/Duce:

You bet that the next roll will be a 3.  The payout is 15 to 1.  Since this has the same probability as an 11 and the same payout, the expected value will also be -11.11, that is, on a \$100 bet, you will lose on average \$11.11.

Two:

You bet that the next roll will be a 2.  The payout is 30 to 1.  Compute

P(Winning Two)  =  1/36

P(Losing Eleven)  =  1 - 1/36  =  35/36

The expected value for a \$100 wager is

(3000)(1/36) + (-100)(35/36)  =  -13.89

Hence on a \$100 bet, you will lose on the average \$13.89.

Twelve:

You bet that the next roll will be a 12.  The payout is 30 to 1.  Since this has the same probability as a 2 and the same payout, the expected value will also be -13.89, that is, on a \$100 bet, you will lose on average \$13.89.

Horn Bet:

The Horn Bet is evenly divided between the number two, three, eleven, and twelve.  If three or eleven rolls, the payout is 3 to 1.  If two or twelve rolls, then the payoff is 6.75 to 1.

First we compute the probability of a three or eleven:

P(Three of Eleven)  =  4/36  =  1/9

Next the probability of a two or a twelve is

P(Two or Twelve)  =  2/36  =  1/18

The probability of losing is

P(Losing)  =  1 - 4/36 - 2/36  =  30/36  =  5/6

The expected value for a \$100 wager is

(300)(1/9) + (675)(1/18) + (-100)(5/6)  =  -12.50

Hence on a \$100 bet, you will lose on the average \$13.89.

Hard Ways

A hard way bet wins if the roll is a chosen pair (4, 6, 8, or 10) and loses if it is a seven or a non-pair with the chosen value.  For example, a hard way four means you win if two twos land and you lose if a seven, or a three and a one land.  Below are the calculations of the six hard ways bets.

Hard Ways 4 -  The payback is 7 to 1.  Since there are three ways of getting a 4 and six ways of getting a 7, the probability of winning is

P(Winning Hard Ways 4)  =  1/9

and the probability of losing is

P(Losing Hard Ways 4)  =  1 - 1/9  =  8/9

The expected value for a \$100 wager is

(700)(1/9) + (-100)(8/9)  =  -11.11

Hence on a \$100 bet, you will lose on the average \$11.11.

Hard Ways 10 - The payback is also 7 to 1.  Since the expected value is the same as a Hard Ways 4, that is on a \$100 bet, you can expect to lose on average \$11.11.

Hard Ways 6 - The payback is 9 to 1.  Since there are five ways of getting a 4 and six ways of getting a 7, the probability of winning is

P(Winning Hard Ways 6)  =  1/11

and the probability of losing is

P(Losing Hard Ways 4)  =  1 - 1/11  =  10/11

The expected value for a \$100 wager is

(900)(1/11) + (-100)(10/11)  =  -9.09

Hence on a \$100 bet, you will lose on the average \$9.09.

Hard Ways 8 - The payback is also 9 to 1.  Since the expected value is the same as a Hard Ways 6, that is on a \$100 bet, you can expect to lose on average \$9.09.

Place Bets

A place bet can be made on any of the numbers 4, 5, 6, 8, 9, or 10.  If the dice sum to the number chosen before the 7 is rolled, then the player wins.  Otherwise the player loses.  Below are the calculations:

4:  The payback is 9 to 5.  Since there are three ways of getting a 4 and six ways of getting a 7, the probability of winning the 4 place bet is

P(Winning 4 Place Bet)  =  3/9  =  1/3

and the probability of losing is

P(Losing 4 Place Bet)  =  1 - 1/3  =  2/3

The expected value for a \$100 wager is

(180)(1/3) + (-100)(2/3)  =  -6.67

Hence on a \$100 bet, you can expect to lose on average \$6.67.

10:  Since the payback and probabilities are the same as for the 4 Place Bet, the expected value is also the same.  That is, on a \$100 wager you can expect to lose on average \$6.67.

5:  The payback is 7 to 5.  Since there are four ways of getting a 5 and six ways of getting a 7, the probability of winning the 5 place bet is

P(Winning 5 Place Bet)  =  4/10  =  2/5

and the probability of losing is

P(Losing 5 Place Bet)  =  1 - 2/5  =  3/5

The expected value for a \$100 wager is

(140)(2/5) + (-100)(3/5)  =  -4.00

Hence on a \$100 bet, you can expect to lose on average \$4.00.

9:  Since the payback and probabilities are the same as for the 5 Place Bet, the expected value is also the same.  That is, on a \$100 wager you can expect to lose on average \$4.00.

6:  The payback is 7 to 6.  Since there are five ways of getting a 6 and six ways of getting a 7, the probability of winning the 6 place bet is

P(Winning 6 Place Bet)  =  5/11

and the probability of losing is

P(Losing 6 Place Bet)  =  1 - 5/11  =  6/11

The expected value for a \$100 wager is

(116.67)(5/11) + (-100)(6/11)  =  -1.52

Hence on a \$100 bet, you can expect to lose on average \$1.52.

8:  Since the payback and probabilities are the same as for the 6 Place Bet, the expected value is also the same.  That is, on a \$100 wager you can expect to lose on average \$1.52.

These are like place bets (as the player and as the house), but pays true odds.  The house automatically takes a 5% commission.

Conclusions

The tables below show the expected values for each of the bets in craps.  It is clear that there are no bets that beat the house on the average.  It is also evident that the Pass Line, Come Line, Don't Pass, and Don't Come bets are superior to the other bets, with only the Place Bet on a 6 or 8 coming close.

General Bets

 Bet Pass Line Come Line Don't Pass Don't Come Taking Odds Laying Odds Field Bet Big 6 Big 8 Loss \$1.41 \$1.41 \$1.40 \$1.40 0 0 \$2.78 \$9.09 \$9.09

Proposition Bets

 Bet Any Craps Any 7 Eleven Ace/Duce Two Twelve Horn Bet Loss \$11.11 \$16.67 \$11.11 \$11.11 13.89 13.89 \$13.89

Hard Ways

 Bet 4 6 8 10 Loss \$11.11 \$9.09 \$9.09 \$11.11

Place Bets

 Bet 4 5 6 8 9 10 Loss \$6.67 \$4.00 \$1.52 \$1.52 4 6.67

Always decide how much you are willing to spend before you begin.  Never go beyond this number.  If you find yourself pulling out more cash to get back what you lost, then you have a gambling problem and should call the National Council on Problem Gambling at 1-800-522-4700.

Back to the Gambling and Math Page

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1

 Sum Freq 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1