Probability, Gambling, and Common Sense

By | April 23, 2021

Recreational gamblers don’t need to understand probability that well. They probably already have a general idea that the casino is always going to win, and they’re okay with that. After all, they’re just paying for entertainment.

Common sense tells most people that.

But you will lose less money in the long run if you have a general understanding of probability beyond what common sense tells you.

Also, I’m convinced that gambling is more fun when you understand the probability underlying everything that’s happening.

This post examines probability as it relates to both gambling and where common sense begins and ends when it comes to probability.

A Common-Sense Explanation of Probability

Probability measures how often something happens given a long time frame. Gambling writers refer to this as “the long run,” which is not to be confused with the album of the same name by the Eagles.

Here are some examples:

If you flip a normal coin repeatedly all days, you’ll expect it to land on heads 1/2 of the time.

If you take a deck of cards and draw a card at random repeatedly all day, you’ll expect to get a spade 1/4 of the time.

If you roll a standard 6-sided die thousands of time, 1/6 of the time is how often you expect to see that die land on 6.

Math Equation

That’s as common-sense a way of describing the probability of an event as I can imagine. Most people understand this concept intuitively.

The math starts getting more interesting when you start thinking about some other aspects of probability.

For one thing, probability is always a fraction – a number between 0 and 1.

Events that are impossible have a probability of 0, and events that are certain have a probability of 1.

Different Ways of Expressing Fractions Applies to Expressing Probabilities

You don’t have to use a fraction to represent an event’s probability. If you paid attention in math class, you already know that fractions can be converted both to decimals. And decimals can be converted to percentages.

So, the probability of getting heads when flipping a coin is 0.5.

The probability of drawing a spade from a deck of cards is 0.25.

The probability of rolling a 6 on a 6-side die is 0.1667.

You can convert those decimals to percentages must by multiplying by 100 or by moving the decimal point to the right by 2 digits. So, you wind up with 50%, 25%, and 16.67%.

All these numbers are common sense for most people, too.

Some of this involves doing some basic math in our head. The real probability of drawing a spade from a deck of cards is 13 divided by 52, but you can reduce that fraction easily in your head.

You can also restate these probabilities as odds. For example, the odds of drawing a spade out of a deck of cards is 3 to 1, and the odds of rolling a 6 on a die are 5 to 1.

And odds are important to a gambler.

Why Using Odds to Express Probability Is So Important in Gambling

Even though percentages are more intuitive for average people, gamblers who’ve been at it for a while and know what they’re doing often find that stating a probability in terms of odds is more useful. During calculations, you’ll often use fractional probabilities to do the math and convert them back into odds later.

Odds compare the number of ways you can lose with the number of ways you can win. It’s a ratio of wins to losses.

Let’s say you have a situation where you are going to win 3/10 of the time. The odds are 7 to 3 – you have 7 ways to lose and 3 ways to win.

You subtract the number of ways to win from the total possible number of outcomes to get the number of ways to lose, then you compare the 2 with each other.

Odds

This is important because odds are also used to explain how much you get paid out when you win a bet. Many bets pay out at even money – 1 to 1 odds, but other bets might pay out at 3 to 2 odds or 2 to 1 odds.

If a bet pays out at better odds than the odds of winning, you’re in a profitable situation.

If a bet pays out at worse odds than the odds of winning, you’re in an unprofitable situation.

In almost all casino game situations, the odds of winning are worst than the payout odds. This is how the casino makes its money.

An Example of the Casino’s House Edge Using Odds and Probability

Let’s say you’re playing a game in the casino where you’re guessing a number between 1 and 10. The dealer has a random number generator that provides each number with an equal probability of occurring.

The odds of winning that bet are 9 to 1, but the casino pays off at 8 to 1 when you win.

Can you see how the casino is going to win money from you in the long run?

Over 10 bets, you’ll lose an average of 9 bets, but you’ll only win 8 bets on the 1 successful guess. The casino will win an extra bet from you every 10 bets, which means that the house edge on this bet is 10%.

This is how all casino games work, but the math behind these calculations might be more involved. Obviously when you’re dealing with cards and multiple dice and roulette wheels with 38 numbers on them, the math gets more involved.

But the principle remains the same – the bets pay off at lower odds than the odds of winning, and that’s how the casinos stay so profitable.

Probabilities for Multiple Events

You’ll often come across situations where you’ll want to know the probability that multiple events will happen. Usually, you can just multiply the probabilities of each event by each other to get the probability that both will happen.

Here’s an example:

Suppose you want to know the probability of rolling a 12 on 2 dice. To accomplish this, you must roll a 6 on the first die and a 6 on the second die.

The probability is 1/6 X 1/6, or 1/36.

In odds terms, that’s 35 to 1.

But that only applies to “independent” events.

Sometimes the probability of a 2nd event will change based on the first event. You must adjust the probability of the 2nd event accordingly before you do your calculations.

Here’s an example of that:

Suppose you want to know the probability of being dealt a pair of aces as your hole cards in Texas hold’em?

The probability of the first card being an ace is 1/13, right? You have 4 aces and 52 total cards.

But once you get that first ace, you only have 3 aces left in the deck out of 51 cards. That changes the probability for the 2nd card from 1/13 to 1/17.

So, the probability of getting a pair of aces as your hole cards are 1/13 X 1/17, or 1/221.

That’s 220 to 1 odds.

You can carry this calculation out as far as you need to.

Probabilities in Sporting Events

Suppose you want to know the probability that the Dallas Cowboys will play the Buffalo Bills in the Super Bowl. To determine that probability, you’d multiply the probability that the Dallas Cowboys will win the NFC by the probability that the Buffalo Bills will win the AFC.

The odds of Dallas winning the NFC aren’t affected by if Buffalo wins the AFC.

So, the math is straightforward enough once you estimate the odds of each team winning their conference.

For simplicity’s sake, let’s just assume that each team has a 1/16 probability of winning their conference. The odds of them playing each other in the Super Bowl are, therefore, 1/16 X 1/16, or 1/256 – or 255 to 1.

What If You Want to Know the Odds that at Least One Thing Will Happen?

Instead of calculating the probability of event A AND event B happening, you want to know the probability that event A OR event B will happen.

How does the math work then?

Usually, you just add the 2 probabilities together.

For example, if you take our earlier example of the Dallas Cowboys and the Buffalo Bills, and you want to calculate the probability that at least one of them will play in the Super Bowl, you just add the probabilities together.

In this case, 1/16 + 1/16 is 2/16, or 1/8.

The odds are 7 to 1.

But this can be an oversimplification. A lot of times, a more accurate calculation involves solving for how many ways a bet can lose and subtracting that from 100%.

Conclusion

Probability seems like common-sense at first, but there’s more to it than that.

Entire textbooks have been written about probability, but most gamblers don’t need to read or study an entire textbook unless they’re really serious about getting an edge while gambling.

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