# Ask the Wizard #1

From your Let it Ride section, you do not list a low pair as a good hand to "let it ride" on. Just how bad of a bet would it be to go contrary to your advice?

Kevin

Letting it ride on a low pair (9's or less) is definitely a bad bet. The house edge on a low pair with three cards is 6.37%. With four cards the house edge jumps to 45.83%. So don't be tempted to let it ride on low pairs.

If a slot machine had five reels, and the probability of getting a cherry were the same on each reel, what would be the probability of getting any specified number of cherries on a spin?

Jay

Let's let p be the probability of getting a cherry on any given reel and n be the number of cherries on the payline. The probability of getting n cherries is combin(5,n) * pn * (1-p)^{5-n}. Combin(5,n) denotes the number of ways that n cherries can appear on five different reels. In particular, combin(5,0)=1, combin(5,1)=5, combin(5,2)=10, combin(5,3)=10, combin(5,4)=5, and combin(5,5)=1. This function can be used directly in Excel and is explained in more detail in my section on probabilities in {poker}. However, to take a specific example, if the probability were 5% of getting a cherry on any given reel then probability of getting 3 cherries would be 10 * .053 * .952 = 0.001128125 .

If one bet on two columns in roulette the probability of winning would be 24/38, or 63%. This seems like a winning strategy to me, what is your opinion?

Anonymous

In roulette any bet or combination of bets carries a high house edge. The more likely you are to win the more you will have to risk relative to the reward. If you do this 10 times the probability of showing a profit is 46.42%. At 100 times the probability drops to 24.6%.

I was playing baccarat online and out of 75 hands the banker won 52 and the player 23. This is a difference of 29, what is the probability of that happening?

Jon from Danville, New Hampshire, USA

First, I'm going to assume that you are not counting ties. In other words, you mean 75 bets resolved. It would be very unlikely to go 75 hands without a tie. The expected number of banker wins out of 75 bets resolved is 38.00913745. The standard deviation is the square root of the product of 75, the probability of a banker win, and the probability of a player win. The probability of a banker win, given that there wasn't a tie, is 0.506788499 and the probability of a player win is 0.493211501 . The standard deviation is thus 4.329727904 . Then you'll have to make a half point correction for a binomial distribution and look up the Z statistic in a standard normal table (this step is left to the reader). The final answer is that the probability of the banker getting 52 or more wins is .0009. Your question also allowed for the possibility of the banker winning 23 or fewer times (also a difference of 29 more more) which has a probability of .0004 . So the final answer is that the probability of a difference of 29 or more is .0013, or 1 in 769.