♠Ask the Wizard: Craps (Dice, and Rolling Dice)

This is about controlling the dice at Craps. You previously discussed the Stanford Wong Experiment, stating, "The terms of the bet were whether precision shooters could roll fewer than 79.5 sevens in 500 rolls of the dice. The expected number in a random game would be 83.33. The probability of rolling 79 or fewer sevens in 500 random rolls is 32.66%.... The probability of rolling 74 or fewer sevens in 500 random rolls is 14.41%."

The question I have about this bet is that 14.41% still isn't "statistically significant" [ i.e. p < 0.05 ] , which is usually taken to mean greater than two Standard Deviations from the Mean -- or a probability of less than a *combined* 5% of the event happening randomly on EITHER end of the series.

How many Sevens would have to be rolled in 500 rolls before you could say that there is a less than 2.5% chance that the outcome was entirely random (i.e. that the outcome was statistically significant) ?

Many Thanks & BTW , yours is ABSOLUTELY the BEST web site on the subject of gambling odds & probabilities that I've found .... keep up the good work !!! - Plexus from Warwick, Rhode Island

Thank you for the kind words. You should not state the probability that the throws were non-random is p. The way it should be phrased is the probability that a random game would produce such a result is p. Nobody expected 500 rolls to prove or disprove anything. It wasn’t I who set the line at 79.5 sevens, but I doubt it was chosen to be statistically significant; but rather, I suspect the it was a point at which both parties would agree to the bet.

The 2.5% level of significance is 1.96 standard deviations from expectations. This can be found with the formula =normsinv(0.025) in Excel. The standard deviation of 500 rolls is sqr(500*(1/6)*(5/6)) = 8.333. So 1.96 standard deviations is 1.96 * 8.333 = 16.333 rolls south of expectations. The expected number of sevens in 500 throws is 500*(1/6) = 83.333. So 1.96 standard deviations south of that is 83.333 – 16.333 = 67. Checking this using the binomial distribution, the exact probability of 67 or fewer sevens is 2.627%.

July 17, 2007

Normal Craps are not allowed in California. Here many casinos are using cards to act as dice, using A,2,3,4,5,6 to act as the 6 sides of the dice. I would assume by using multiple decks it would alter the odds. (i.e 4 decks = 16 aces, 16 2's, ect.) Does this favor the house as in blackjack... or does this favor the player? The player could bet at higher or lower numbers based on the half of the cards out of the shoe before a shuffle(assuming a mid shoe shuffle). -- Joe from Eureka, Ca

You're correct, dice alone can not determine the outcome in craps. There are various ways of using cards in place of dice and still have the odds exactly the same. One way is to use two separate decks, thus there is no effect of removal. Another way is to have a 7-card deck, featuring the numbers 1 to 6, plus a seventh "double" card. The first card drawn can never be the double card. If it is then it is put back in and the process repeats from the beginning. If the double card is drawn second then it counts as whatever the first number drawn was. Regardless of how the casino does it I have never seen hard evidence of a case where the odds were different than if two dice were used. So I think you are omitting something from the rules. March 13, 2006

Do you believe the toss of the dice at a casino craps table is truly random as a RNG would be, or are there good shooters and bad shooters either thru dice "mechanics" or plain sloppy throwing (short throws as an example), if real world Casino Craps is not truly random , how would I take advantage of this ?

Are dice truly unbiased? It seems like the sides with the larger numbers which have more holes would be lighter than the sides with the smaller numbers and less holes. This seems to suggest that the heavier sides would more likely land face down with the larger numbers more likely landing face up. I can imagine a craps system that could try to exploit this principle, but I wonder if it would really work. What do you think? - Samuel from the USA

I recently learned some information about dicesetting strategies in craps. Some believe that you can set the dice a certain way before the throw, and by keeping the roll of the dice to just one axis of rotation, you can have fewer possible sevens with certain dice sets. I wanted to know if there is any truth to this or is it just a fallacy. - Duane Vick from Las Vegas, USA

Just wondering about your opinion of altering the frequency table in craps by pre-setting the dice. - Thom from San Francisco, California

Q: Do you believe that "wishful thinking" on behalf of the players can affect the outcome of a game. Note that I'm not concerned with the SIZE of the effect, just your philosophical opinion. Also, do you think that the manner in which a player tosses the dice in craps can cause a bias (good or bad) in the outcome. As always, your site is AWESOME. Thanks - Ted Formeza of Perth Amboy, New Jersey

Q: I was wondering how to alter dice in the game of craps so it hits 7 or 11 every time can you help thank you. -- d from Monterey, California