Gambling and Odds

Half of everything is luck. And the other half? Fate.

Back to Math and Logic

 

Common Misconceptions

Dependent vs Independent Events

 

  • If you flip a coin 50 times and it comes up heads each time, what are the odds it will come up heads on the next coin flip?

50%. The key insight here is that each coin flip is an independent event. Regardless of what the coin did before, the odds are exactly the same every time. Similarly, your odds of winning or losing a hand of poker, slots, whatever else isn't tied to what happened in the last game.

  • You're playing five card draw and the dealer gives you four spades and a club. You draw one card, what are the odds of getting a spade?

In this case since all your cards are coming from a fixed pool of cards, the odds are 9 out of 47, there are nine spades left, and 47 cards you don't know the value of.

 

Doubling up

 

Originally known as the Martingale System, the basic premise is that you double up every time you lose a bet. For example, you can start with betting a dollar and if you lose you bet two dollars on the next round. If you win that round you're ahead a dollar, if not you bet four dollars on the next round. If you win that round you're still ahead a dollar, if not you keep doubling up until you've made back your initial wager or you go bankrupt.

The system essentially works like a reverse lottery. Most of the time you can expect to win small amounts of money, but if you lose, you lose everything. The system usually lasts until the player runs out of money or decides to cut his losses and quit. It's important to note that this system is still a losing bet in the long run, and casinos are able to make money off of players who use these systems since it always pays off for them in the long run. The trick to this system is that it can be a winning bet in the short run.

For example, let's say you have a bankroll of $1024 dollars. You start out betting one dollar and double that amount every time you lose to make back your wager, meaning you can double up 10 times before you lose. You're playing blackjack, so you have roughly a .49 chance of winning any particular bet.

The following table shows the odds of winning a specific number of bets without going bankrupt, meaning you can play 50 rounds and still have a 94% chance of winning 50 dollars, although your expected value on those bets is still negative.

Total Bets Odds of winning bets Expected Value

0.9988

-0.22

25

0.9706

-5.77

50 

0.9421

-12.09

100 

0.8877

-26.21

250

0.7424

-78.10

350 

0.6590

-118.41

500

0.5512

-183.89

598*

0.4905

-228.38

1024

0.2953

-419.19

 

*Note that from this point onwards, using the system is worse than just placing a flat bet, since with a flat bet of any amount you have a .49 chance of winning, with an expected value of -.02*YourBet. For example, if you just bet 600, your expected value is -12.

Now, most casinos have a minimum and maximum bet which prevents you from doubling up more than six times. In this type of a system, where the number of times you can double up is six:  

Total Bets Odds of winning bets Expected Value

0.9824

-0.14

5

0.9150

-0.86

10

0.8373

-2.03

20 

0.7011

-5.10

30

0.5870

-8.81

40*

0.4915

-12.87

64

0.3210

-22.90

 

*Note that again, from this point onward, using the system is worse than just placing a flat bet.

And just for good measure, here's a table that shows at what point you still have a 75% chance of winning all your bets for a specific size of your stake, and at what point you still have a 49% chance of winning. Note that the size of your stake here represents the number of times you're able to double up, 5 through 10 in this case.

 

Total Size of your Stake  Still good odds Still advantageous

32 

8 bets

20 bets

64

16 bets

40 bets

128

31 bets

79 bets

256 

62 bets

155 bets

512

123 bets

305 bets

1024

241 bets

598 bets

 

In other words, you can expect to make back between one fourth and one fifth of your initial stake with decent odds if you don't mind a 1/4 chance that you'll lose the whole thing.

And here's one last table aggregating the data further, showing how much of your initial stake you can hope to win for a particular margin of risk. (assuming you can double up 10 times. Note that in some cases you're better off making your initial bid a higher percentage of your stake, the table also notes what proportions you should divide your stake into to achieve the following values):

Acceptable Margin of Risk Expected percentage of stake that can be won

95%

4.3% (doubling 8 times)

90%

9.38% (doubling 5 times)

80%

18.75% (doubling 4 times)

70%

31.25% (doubling 4 times)

60%

43.75% (doubling 4 times)

50%

60.94% (doubling 6 times)

 

Note that the 50% risk margin is even worse than it appears, because in a straight bet in which you risk the same amount of capital, you can win 100% of your initial stake.

Streaks

Poker

 

There are four variables to consider in whether you should stay in for a hand of poker.

  • P: The odds that you are going to win with your current hand
  • Pot: the size of the current pot
  • B: the amount you are required to bet to stay in
  • N: the number of other players in on the call

Note that this analysis doesn't take into account bluffing or anticipating the strategy of other players, which may be used to change your estimate of P, or the odds that they're holding good hands. It's also possible to win by bluffing yourself, which further skews P. Being able to bluff well and anticipate other players is computationally and mathematically infeasible to have a perfect strategy for, so this page will just discuss the straight odds of winning.

In general, the amount of money you can expect to win from a particular play is:

  • The odds of winning* (The size of the pot + the size of the remaining bets from the other players) - The odds of losing * the size of your remaining bets
  • Or: P*(Pot+N*B) - (1-P)*B

If that value is positive, it's worth your money to stay in, it means that if you play that same situation enough times you're guaranteed to come out ahead in the long run. If that value is negative, it means that if you play that same situation enough times you'll come out behind. And if that value is zero, you'll break even.

There are two crucial decision making points for a game of Poker:

  • Whether or not to stay in to see more cards, where the odds of winning are affected by what hand you think you can get
  • Whether or not to stay in until the end, where the odds of winning or affected by what hands you expect the other players to have

In the case of a game like five-card draw there's also the decision to be made of which cards to keep and which to discard in order to get the best hand possible, but that decision is fairly static, and not generally dependent on the state of the game.

 

Planned randomness

To simulate a coin flip, look down at the second hand of your watch. If it's even, call it heads, if it's odd, call it tails.