Figuring the Odds

2 July 2023

By John Grochowski

QUESTION: I've read the odds are the same on every play no matter what has happened before. If roulette has stopped on black five times in a row, the odds on another black are still the same. If a craps shooter has rolled five sevens in a row, odds are the same. Even slot machines, odds are the same even after jackpots or winning streaks.

I can't wrap my head around how this is possible. Doesn't there have to be a streak with no blacks or no 7s to get the percentages back in line? The odds against six black numbers in a row have to be really long.

ANSWER: The odds are stacked against any long streak before it starts, but the odds against any single win in the streak are just the odds of winning once.

For easy math, let's use a coin flip as an example. The chances of tails turning up on one flip are 1 in 2 -- one tail out of two possible outcomes. Chances of two in a row are 1 in 4, derived by multiplying 1 in 2 by 1 in 2. Six in a row? That's a 1 in 64 shot.

But imagine you've already flipped five tails in a row and you're going for the sixth. you've already done the hard part. Winning five in a row has already taken care of a 1 in 32 chance. What's left to make it six in a row is one more win with a 1 in 2 chance -- the same as on any other flip.

There is nothing that would make the odds against tails any longer or shorter on any one flip. The coin doesn't know what's gone before.

The odds vary with different bets. The chances of a ball in double-zero roulette landing in black aren't 1 in 2, they're 18 in 38, which reduces to 1 in 2.11. The chances of six wins in a row aren't 1 in 64, they're a fraction more than 1 in 88. But once you've won five in a row, the chances of a sixth are 1 in 2.11.

In craps, six 7s in a row is a 1 in 46,656 long shot. But once there have been five in a row, the chance of the sixth is 1 in 6.

Casinos are in business for the long haul. They understand that over hundreds of thousands or millions of plays, hot streaks, cold streaks, and jackpots will fade into statistical insignificance.


QUESTION: I strongly suspect a casino of removing high cards from blackjack decks. What should I do about it? You should warn other players.

ANSWER: Your move should be to list the reasons for your suspicions and contact your state gaming board. If you continue to play there and more evidence mounts, pass that on to the gaming board, too. And if you know other players who've had the same experience, urge them to make their own reports. But if I thought a casino was cheating, I wouldn't play there.

Without knowing your evidence, the likelihood is that there's nothing wrong with the game and that you've just hit a bad streak. If your state is like most, whenever fresh decks are brought to the table they must be opened and spread out so players and security cameras can see all cards are there.

I'm afraid I can't pass on a warning. I'd need really strong, verifiable evidence something was wrong with a game before I tarred a casino's reputation. One player's suspicions won't do it.