Ask the Wizard #83

Assuming you had an equal chance at getting each horse then the probability of winning would be 1 in 17, regardless of how the odds on each horse were distributed. The player’s expected return can be expressed as (1/17)*15.3 + (16/17)*-1 = -0.0412. In other words the house edge is 4.12%. However there is another simple formula you can use. If a = actual odds paid and f = fair odds for bet then the house edge is (f-a)/(f+1). In this case the fair odds are 16 to 1. So the house edge is (16-15.3)/(16+1) = 0.7/17 = 4.12%.

I addressed robot players in the September 20, 2001 column. Assuming you could create a robot then it should do well against a game with over 100% return. However I would recommend programming it to play at a human speed and reasonable sitting times. Some online casinos have been known to blame players for using robots even when they weren’t, as an excuse not to pay in my opinion.

I get challenged a lot on my support for come bets with odds. Those against come bets are always quick to say a number has to hit twice with a come bet and only once with a place or put bet. That is not a good way to look at it. First of all with come bets you have a 8/36=22.22% chance of winning on the first roll as opposed to only 11.11% of losing. You get no such advantage on the first roll of a place or bet bet. Even if you do roll a point number it can be any number. In other words there are six point numbers that can win on a come bet, and only one on a place or put bet. Ultimately the reason the come bet with odds beats place or put bets is the house edge is less.

Let’s assume six decks of cards and the player always takes a third card (whether by hitting or splitting). The number of ways to draw 3 suited sevens is the number of suits (4) times the number of ways to choose 3 out of 6 sevens of that suit in the shoe. In other words 4×combin(6,3)=4×20=80. The number of ways to draw 3 colored sevens, including 3 suited sevens, is the number of colors times the number of ways to pick 3 out of the 12 sevens in the shoe of that color, or 2×combin(12,3)=2×220=440. The number of ways to draw any 3 sevens, including 3 colored and suited sevens, is the number of ways to pick 3 cards out of the 24 sevens in the shoe, or combin(24,3)=2024. The total number of combinations for any 3 cards out of 312 is combin(312,3)=5013320. So the probability of 3 suited sevens is 80/5013320=0.000015957. The probability of three colored, but not suited, sevens is (440-80)/5013320=0.0000718. The probability of three sevens of mixed colors is (2024-440)/5013320= 0.00031596.

Yes, I calculate blackjack odds using a combinatorial approach, analyzing every possible ways the player and dealer cards can come out, taking the greatest expected value at every decision point. This is harder to program than a simulation but I feel is more elegant and a nice challenge in recursive programming. However I still respect my peers to do simulations. With today’s computers it doesn’t take long to run off a billion bets, which gets very close to the optimal strategy return.