Simplified Mathematics: Probabilities of casino games.
A lot of people don’t consider themselves to be good at math. What so many don’t realize is that most mathematical equations – even the dreaded fractions and percentages – don’t have to involve difficult math. I suppose it really depends on what level of math a person considers their skills to go downhill.
If you can do basic addition, subtraction, multiplication and division – we’re talking grade-school stuff – you have what it takes to do something so seemingly complicated as calculating probabilities in gambling.
I’m going to take this slowly, step by step, to show just how easy it is. Relax and take a deep breath – you got this!
Probabilities of Flipping a Coin
This is as easy as it gets, but don’t skip ahead. I’m starting here for a reason. It will help you understand bigger calculations later on.
Probabilities are properly expressed as a number between 0 and 1. When flipping a coin, there are only two possible outcomes – heads or tails. The probability of one of those outcomes happening is 0.5. Flipping heads is 0.5, and flipping tails is 0.5. Put them together and you get a whole 1.
That decimal can also be expressed as a percentage (50%), a fraction (1/2), or as odds of 1 to 1 (1 chance of this happening, or 1 chance of that happening).
Moving right along…
Probability of Drawing a Card Suit
Let’s expand the equation to something where one of four outcomes is possible. Let’s say you have four playing cards, one of each suit. What are the odds you will draw a club?
There are four possible outcomes, and we’re looking for just one of them. We divide the probability of all outcomes (1) by the number of possible outcomes (4), and we get 1/4 = 0.25. Or, we can just display this as the fraction of 1/4 (a one in four chance). Or, we can convert that to a percentage – a 25% chance. As odds, it would be expressed 3 to 1 – the number of undesired outcomes (3) “to” the number of desired outcomes (1).
Are you still with me? Good, let’s get back to the coin toss for our next lesson.
Probability of Flipping 2 Heads in a Row
Now things get more complicated, right? Not really. To get the odds of a specific outcome occurring twice, you simply multiply the odds of that outcome happening once, by the odds of that outcome happening once (again).
We know already that the odds of flipping a coin and it landing on heads is 0.5. So we multiply that integer by itself. 0.5 * 0.5 = 0.25
So the probability is 0.25, or a 25% chance (1/4, or 3 to 1) that heads will appear twice in a row.
Perhaps the math makes sense to you, but if not, here’s a really simple way to explain why that equation is correct.
Let’s think about how many things could possibly happen in two flips of a coin. The only possible outcomes are:
One Heads, One Tails
One Tails, One Heads
As you can see, there are only four possible outcomes. As we learned in the card suits example above, four outcomes means a 1 in 4 chance, or 0.25 probability.
The Probability of Drawing an Ace
Things get more complicated now, because we have 52 possibilities in a standard deck (no jokers), and four of the are an Ace. Therefore the odds of drawing an ace from a full deck are 4 in 52, or 1 in 13 (52 / 4 = 13). As a percentage, we can calculate 1 / 13 = 0.0769, which translates to 7.7%.
It may not be easy to come to 7.7% odds in your head, but the 1 in 13 should be obvious. There are 13 different cards per card, four of each. Everyone who plays cards knows there are 52 (minus jokers) in a deck.
2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace
So if there are four of each card (4 in 52), there has to be 1 for every 13 (1 in 13). You don’t need a calculator to come to this result. It’s simple logic. And as an estimate, it’s easily apparent that a 1 in 13 chance is going to be under 10%.
Probabilities for Blackjack Card Counting Purposes
Card counting isn’t nearly so difficult as movies like “21” make it sound. Sure, there are more intricate counting systems, but a basic count is almost just as effective, and is incredibly easy to manage. But I’m not here to teach you how to count cards. I’m going to teach you casino math – in this case, how to estimate the remaining Aces in a blackjack game.
What you’ll need to know is a) how many decks of cards are in the shoe, b) how many Aces have already been played, and c) how many total cards have been played.
Let’s say it’s a 6-deck shoe. 6 decks of 52 is 312 cards.
6 * 52 = 312 Cards
If you would rather estimate, that’s about 6 * 50 = 300.
There are 4 Aces per deck, so 24 Aces in the shoe.
6 * 4 = 24 Aces
With this information, we know that before any cards are dealt, there is a 24 in 312 (which breaks down to the same 1 in 13 above) chance of being dealt an Ace.
For sake of exemplification, let’s say you’ve been playing for a bit, you’re about ¼ of the way into the shoe with 78 cards seen. That leaves (312 – 78) 234 cards in the deck. If you’ve witnessed the deal of 2 Aces in that time, there are (24 – 2) 22 Aces left. Now the odds change…
22 in 234 chance of being dealt an Ace.
22 / 234 = 0.094, or 9.4% odds of being dealt an Ace
234 / 22 = 10.63, or a 1 in 10.6 chance
Again, it’s easy enough to estimate. Round 234 to 200 and 22 to 20. 200 / 20 = 1 in 10 (10%) chance, which is really close to the exact numbers above.
Probabilities in Roulette
There are two types of roulette wheels – American and European. For the purpose of this guide, I’ll be referring to a European roulette wheel for the simple reason that American roulette has far worse odds. Simply put, don’t play American roulette! Stick to European or French wheels with a single zero (0).
Roulette is not the easiest game to calculate probabilities for, mostly because the numbers involved are not evenly rounded numbers. You may need a calculator for this one. The good news is, the house edge does not change in European roulette. The payout for each wager is set at such a rate that the edge remains a static 2.7% at all times.
Let’s have a closer look…
Single Number Roulette Bets
The wheel has 37 slots, ranging from 0 to 36. So, to define the probability of the ball landing in any single number slot, we must divide 1 by 37.
1/37 = .027, or a 2.7% chance
What does this mean? Odds are, if we placed this bet 37 times, we would win one of them, worthy of a 35 to 1 payout of 36 units (35 + 1, the original bet). Thus probabilities states we would lose 1 bet unit it 37 attempts. So, the 2.7% probability also translates to a 2.7% house edge, wherein the player can expect to lose 2.7% of their bets, and win 97.3% of them (aka the theoretical return to player, or RTP).
Let’s consider a different roulette bet, such as a bet on dozens.
Dozens Roulette Bets
There are three groups of dozens containing 12 numbers each. Betting on the first dozen is a bet on 1-12. So there are 12 numbers that will win, and (37 – 12) 25 that will lose. To get the probabilities, we divide 12/37.
12/37 = 0.3243, or a 32.43% chance
We can easily round that up to a 1/3 chance for the purpose of estimation, but it still leaves a little extra wiggle room for the casino to accommodate it’s house edge.
Since this bet pays 2 to 1 (3 units), we can multiply those numbers to get the RTP.
3 * 32.43 = 97.29
And if we subtract the 97.29% RTP from 100%, it rounds to the same 2.7% house edge.
Even-Money Roulette Bets
What about even-money bets? Any wager on black, red, odd, even, high or low is an even-money bet. It is a wager with 18 ways to win and 19 ways to lose (i.e. 18 out of 37 possibilities), paying 1 to 1 (2 units) for a win.
18 / 37 = 0.4864, or a 48.64% chance to win.
Now to find the RTP and house edge, we multiply the units won (2) by the probability (48.64%).
2 * 48.64 = 97.29%
Again, we come to a 97.3% RTP, and a 2.7 house edge.
Whether it’s a 3-number line bet, 4-number corner bet, or 6-number street bet, the end result is always the same – a 2.7% house edge.
The Difference with French Roulette
There is a slight difference in French Roulette. This game features a special rule known as La Partage. By this rule, any even-money wager that loses due to the winning number being zero (0) will result in half of the bet being lost, and the other half being returned.
For this reason, the house edge on even money bets in French roulette is cut in half, from 2.7% to 1.35%.
2.7 / 2 = 1.35
By this reasoning, any truly educated casino gambler will tell you that the French variant is the most advantageous roulette game for players.