Perfect Math, Real House Edge

25 September 2012

By Jerry Stickman

The house edge for a pass line bet with no odds is approximately 1.41 percent. This is a low house edge compared to other table games and even to other craps bets. The casino extracts its edge in one of two ways – paying the player less than fair odds or by winning more decisions than it loses. The pass line bet gets paid even money, so the house gets its edge by winning more than it loses.

A series of calculations, which I will not show here, determines how many hands win and how many lose. With a pass line bet the house wins 251 compared to the players 244 wins. Out of the total of 495 decisions the casino wins seven more than the player so the house edge is 7/495 which equals approximately 1.41 percent.

Does that mean for every 495 decisions at a craps table the house will win 251 and the player will win 244? Absolutely not. In fact it would be very rare for 495 straight decisions to consist of 251 wins for the house and 244 wins for the player. Sometimes the house will win many more than 251. Other times the player will win significantly more than 244.

There are gamblers who claim the house edge of 1.41 percent for a pass line bet is a hoax because there will not be 244 player wins and 251 house wins in every 495 decisions.

As with any random occurrence, predicting a result is impossible. That is the very definition of random. Flipping a coin has two equally possible outcomes; heads or tails. On average a heads will show one out of two times the coin is flipped. But flipping a fair coin is a random event. Even though a head or a tail is an equally likely result, it doesn’t mean that if a head shows the next flip must be a tail. If that were the case the coin flip would not be random.

Suppose you have a decahedron (ten-sided polyhedron) with the sides numbered 0 through 9. If the decahedron were perfect, each side would have an equal chance of showing when the decahedron was thrown. On average the 0 would appear once every 10 throws, the 1 would appear once every 10 throws, as would the 2, 3, 4, 5, 6, 7, 8, and 9. Throwing the decahedron is a random event. When the 0 shows, it does not mean all the other nine numbers must show before the zero shows again.

Look at the series of 1,000 numbers below:

1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

These numbers are the fractional part of Pi (p). The numbers 0 through 9 are in a random sequence, Notice six 9’s in the fifth row and nine numbers down. In a random event anything can happen. The next number however is not predictable.

In a perfect world, the above 1,000 digit series would contain 100 0’s, 100 1’s, 100 2’s, etc. I did not take the time to count each number but I did count the 0’s, 1’s and 2’s. The totals are: 0 = 90, 1 = 116, 2 = 98. So they are close to 100 but not equal to 100. As thousands and thousands more digits are produced each digit 0 through 9 will come closer and closer to being one-tenth of the total digits. Such is the nature of the math.

Going back to our pass line discussion, each group of 495 decisions will probably not have 251 house wins and 244 player wins, but over the course of millions of decisions the wins and losses will average 251 wins for the house and 244 wins for the player for every 495 decisions.

Math counts the long term which is the only thing that makes sense in random events. Just because each 495 decisions does not have exactly 251 house wins and 244 player wins doesn’t mean the house edge in not 1.41 percent. Random is random (unpredictable) and math is math (consistent).

May all your wins be swift and large and all your losses slow and tiny.

Jerry “Stickman” is an expert in craps, blackjack and video poker and advantage slot machine play. You can contact Jerry “Stickman” at stickmanGTC@aol.com