Throwing Out Ties (Absolute versus Relative Probability)

Gambling Ramblings - Extra Stuff by Peter Griffin

Another common source of disagreement, which occurs even among mathematicians, was highlighted for me by a conversation with the manager of a prospering Las Vegas Casino. He quoted a 1.37% mathematical advantage for the Player bet in Baccarat. This grated against my memory of a 1.24% edge for the house and I demurred. It did not occur to me until later that we both could be considered correct providing that we specified whether we meant percentage advantage per resolved bet (the 1.37%) or per hand dealt (the 1.24%).

The disagreement is quickly resolved by looking at 10,000 Baccarat hands broken down into the ideal categories of 4586 Bank wins, 4462 Player wins and 952 ties. My 1.24% comes from the 4586 - 4462 = 124 units net loss on the Player bet divided by the 10,000 hands played. The 1.37% arises from dividing this 124 unit loss by 9048, the number of hands which don’t result in a tie.

I touched on this point in my first article for C & S in which the 1980 ideal results for the Don’t Pass bet in Craps were categorized as 949 winning bets, 976 losers and 55 standoffs from the bar. The net loss is 27 units which can be summarized either as 27/1980 = 1.36% loss per hand of Craps or 27/1925 = 1.40% loss per resolved bet.

There is no common agreement even among mathematicians about which description is preferable. Professor Stewart Ethier (then at Michigan State University, now at University of Utah) presented an elaborate mathematical argument for throwing out the ties in his paper On the House Advantage presented at the Fifth Conference on Gambling sponsored by the University of Nevada, Reno, at Caesars Tahoe in October of 1981. I tend to disagree with Stew on this, preferring average win per hand played, regardless of ties, on the grounds that the player is not compelled to leave his bet out after a tie occurs: when a bet on Baccarat or on the Don't Pass line in Craps is made we have no guarantee it will be left until resolution of either victory or defeat.