Ask the Wizard #33

It is my understanding that the remaining 47 cards are continuously being shuffled until the player makes a decision what cards to draw. So, the draw cards are not predestined at all. Mathematically speaking, it doesn't make any difference.

Somebody else asked this is a past column. The book Finding the Edge presents a paper titled 'An Analysis of Caribbean Stud Poker' by Peter Griffin and John Gwynn Jr. There they state that if seven players colluded perfectly they would enjoy a 2.3% player advantage. However, they don't state what the edge would be in a five-player game. I suspect that the odds would swing back to the house.

I think what you are referring to is actually called the "Law of Large Numbers." This states that for a random sample of n random variables with mean x, that the sample mean x_n converges to x as the size of the sample approaches infinity. We can think of the outcome of a bet as a random variable. This law tells us that as the number of bets becomes very large the average result will get closer to the house edge.

Thanks for the kind words. You ask a good question for which there is no firm answer. It is more a matter of degree, the more you play the more your results will approach the house edge. I recently replaced my blackjack appendix 4 with some information about the standard deviation which may help. For example this table shows that if you play 10,000 hands of blackjack the probability is 90% of finishing within 192 units where you started after subtracting the expected loss due to the house edge. So in 10,000 hands you are likely to win or lose less than 2% of total money bet due to random variation. However if we go up to one million hands the probability is 90% of an 0.2% variation due to luck. In general the variation in the mean is inversely proportional to the square root of the number of hands you play. All of this assumes flat betting, otherwise the math really gets messy.

Gambling is supposed to be fun, so if you think you get 39.5 cents worth of fun every $5 you bet, then you should play it. That is how much you will give up in the house edge, assuming no other players.

The return is 99.9367%.

I am very confident that any respectable maker of video poker machines makes them fair and accurate. Is it possible there are dishonest machines or chips out there? Sure. I would be interested to read the article you refer to.

The probability of a four of a kind in seven cards is 0.00168067, the probability of a straight flush is 0.00027851. If x is the probability of a four of a kind and y is the probability of a straight flush then the probability you ask for is combin(50,2)*48*x²*y*(1-x-y)⁴⁷. The answer comes out to .0000421845, or 1 in 23,705.