Gambling: Probability, Odds, and Math
It's
my opinion that gambling is a fine activity for healthy, intelligent
people who are able to control themselves and who understand a
little bit about gambling probability. A lot of people have
an intense hatred of math, and so they probably won't enjoy this
discussion of gambling odds and gambling math. My
recommendation to those people is to not gamble. They'll probably
ignore that advice, anyway, which is fine. This site assumes that
all of its users are adults capable of making adult decisions. But
if you want to be an informed gambler and understand a little bit
about the probably and math behind gambling games, then read on.
What Is Probability?
Probability is just a way of measuring how likely it is for a certain outcome to come about. This probability can be an exact number, or it can be an approximate number. The probability of any outcome is a simple math problem. You take the number of ways you can achieve a particular outcome and divide it by the total number of possible outcomes.
That might sound complicated, but it's not. A classic example is the probability of getting heads when you flip a coin. How many ways are there to achieve that outcome? One. How many total possible outcomes are there? Two--heads and tails. What's one divided by 2? It's 0.5.
That number can be represented in multiple ways, but no matter how it's expressed, the probability of anything that might happen is always going to be a number between 0 and 1. Something that is impossible and will never happen has a probability of 0. Something that is certain to happen has a probability of 1.
Here's an example. Suppose you roll a standard six side die. The probability of that die coming up with a seven is 0. There are only six possible outcomes: 1, 2, 3, 4, 5, or 6. Since 7 is never a possible outcome in this situation, it has a probability of 0.
This number can be expressed as a fraction, as a decimal, or as a percentage. Probability can also be expressed in odds format. When expressing probability in an odds format, you take the number of possible of negative outcomes and compare it to the total number of positive outcomes. For example, if you want to roll a 6 on a six-side die, then there are 5 negative outcomes, and 1 positive outcome, so the odds are 5 to 1.
1/2 is the fraction that represents 0.5 in our earlier example. 50% is the same number expressed as a percentage. Even odds, or 1:1 odds, is how that probability would be expressed in odds format.
The total of all the possible probabilities for each outcome will always equal 1.
How Do Probabilities and Math Affect Gambling
All
casino gambling games pay off at odds that are slightly different
from the odds of you winning. The difference between the payout and
the chance of your win is called the house edge, and it's the
percentage that the house keeps over time.
For example, in roulette, you win even money if you bet on black. That represents payout odds of 1:1.
But the true odds of getting black at a roulette table aren't even. An American roulette wheel has 38 slots, and only 18 of them are black. (18 of them are red, and 2 of them are green--the two 0's.) So your odds of landing on black are 18/38, or 0.473, or 47.3%.
So when you make an even money bet at the roulette table, you'll win 47.3% of the time, and you'll lose 52.7% of the time. But you'll win the same amount as you lost, so it doesn't take a rocket scientist to understand that the casino is going to win all your money over the long haul.
The same math applies to the more complicated bets, too, but the calculations take a little bit more work. For example, if you bet on a single number, the bets pays off at 35 to 1. But the actual odds of winning that bet are 37 to 1.
The difference between the odds of winning and the payout is the casino's profit.
A Gambling Probability Example Using Cards in a Card Game
Suppose you want to know the probability of being dealt a spade from a standard deck of 52 cards. There are 13 possible ways of achieving this outcome, out of 52 total possible outcomes. (Each card in the deck is a possible outcome, but only the 13 spades are the winning outcomes in this situation.) You could express that probability as 13/52, but most people would reduce that fraction to 1/4. That can also be expressed as 0.25, 25%, or 3 to 1. If you placed a bet on the likelihood of being dealt a spade, if it paid out at 2 to 1, you'd be placing a negative expectation bet. But if that bet paid out at 5 to 1 instead, you'd have a positive expectation bet.
The math is too complicated to get into here, but the house has an edge over the player in almost every situation. The only exceptions are video poker, sports betting, blackjack, and poker. And only expert players can overcome the house edge on those games, and even then, only in certain situations. For example, some video poker games pay out at slightly over 100% if they're played with perfect strategy. Some sports bettors can find situations where a bet pays out at better odds than are being offered by the sportsbook. Card counters can gain an advantage over the casino, but only if the casino doesn't shuffle the deck too often. And expert poker players can take advantage of the mistakes that amateurs make to operate at a profit, but only if they're good enough (and their opponents are bad enough) to change the odds by so much that it overcomes the rake.
