The Mathematics of Roulette I Understanding Casino Games

For thousands of years, games and puzzles have been an enjoyable and rewarding aspect of human civilization. They tease our brains. They challenge our memories. They strengthen our competitive skills. And whether it’s chess, poker, or Sudoku, most games have this in common: Everything you need to win is rooted in mathematics.

This video is episode two from the series “The Mathematics of Games and Puzzles: From Cards to Sudoku”, Presented by Arthur Benjamin
Learn more about the math of casinoo games at


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35 thoughts on “The Mathematics of Roulette I Understanding Casino Games”

  1. I heard a saying that went something to the effect of "Those who have a system for playing roulette will lose their money systematically." I think it was in a movie. Or was it about Keno?

  2. Winning for the players in casino games is all illusion. Casinos win over you is normal. You win over casinos is only your luck. Consider how many people in the payrolls in the casinos you should know. Welcome to your favorite casino, salvation armies.

  3. You go and only bet red or black, start with $1 or $5..

    You lose, you just double up eventually you will hit a red

    Bet $1, lose, down $1
    Bet $2, lose, down $3
    Bet $4, lose, down $7
    Bet $8, win, up $1
    Pocket that $1 and start over the original $1

    Eventually its gonna hit red

  4. I guess the fundamental point of the house having better odds exists in the existence of the '0' and '00', without which the EV would amount to 0 in all cases. Wonder how this works for European Roulette. Does the house have slightly worse odds in that case?

  5. Professor Benjamin, recently I'm playing a spinning wheel game online. Here's the situation: players bet on four given multiples of the betting amount—2 ,3, 7, 18, and the wheel consists of 37 bars for the pointer to land on, among the 37 bars, there are 18 2s, 12 3s, 5 7s, and 2 18s.
    Apparently, betting on 18 yields the maximum return. However, based on my observation and experience, the frequency for 18 could appear 2 times in a row, and then reappear in 3 to 10 rounds of betting, but eventually I had to patiently wait for 28 to 120 rounds for 18 to pop up again, which is perplexing to put bets on. In one extreme case, the pattern went 18,7,7,18,7,18. On the basis of the laws of probability theory and stochastic processes, what is the safest bet for maximum return when betting on 18 and 7?

  6. Thanks for the $1 /1 number bet. Can you please give me the example of $1 each at 10 numbered play? Would that still calculate at 5.3 cents per dollar? = .53/@ $10. Thanks

  7. In the long run you lose and the casino wins. In the short run you could win some or lose some. Now realize that you come in with a small pool of money, so you probably won't play for the long run. But the casino does play for the long run, and they have a massive pool of money. Casino always wins…

  8. But what if you play 2 rows or play 2 12s instead of just one then your probability would be 24/38 giving you the upper hand if you make the same bet everytime would you come out with a higher edge then the house or am I missing something ? Please let me know ! In my math playing 2 sets of 12 gives you a .26% edge or ev of 2.6 cents profit out of every 1$ you play. ( 1(24/38) + -1 (14/38) = 5/19 or .26315

  9. Some people walk up and will put $100 on red or black with an expected win of 18:38. I contend if that same person put $90 on red or black and $5 on each of the two green numbers they still have $100 on the table but have increased their odds of winning by 2 numbers. Also if they hit one of the green numbers that pays $175 – the $100 bet still clears $75. In the long run this probably would not pay off but for that one time thrill better I like to take insurance. Note if you try this and hit on the green i suggest you put $20 to $30 on split greens for the next spin. Could pay for your weekend trip.

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