The maths behind the cards

Everyone’s got a favourite casino table game. It might be roulette, it might be craps, but for a lot of players it’s blackjack because of the low house edge it has associated with it. Card games in general are pretty popular in the gambling sphere: the World Series of Poker still has a deal to air on ESPN and blackjack frequently features in movies like Rain Man as the game you can beat if you’re smart enough.

But you don’t need to be Dustin Hoffman to work out the probabilities in a game of cards, just read through our quick primer and you’ll be seeing the odds before the card is even dealt.

Step 1: Understanding odds and probabilities

The best way to understand odds and probabilities varies from person to person – some people will find the odds given in sportsbooks, such as 3:1, easy to understand. But it’s surprisingly common for someone to see this and believe that the probability it describes is 33% not 25%.

It’s confusing a probability represented as a fraction with one represented as odds – in simple terms, 3:1 is 3 failures for every 1 success for a total of 4 outcomes. As you succeed 1 time in every 4, the odds are 25%.

We’ve opted to use percentages throughout this guide as we think it’s the most universally understood method of displaying a probability.

Step 2: Understanding card probabilities

With 52 cards in the deck, to calculate the probability, divide 52 by the amount of the type of card in the deck.

Meaning for a single card it’s 1/52, making the probability of it being dealt about 1.9%.

If you’re after a certain value card, then there are 4 in the deck and 13 possible values, so it’s 4/52 or 1/13 which is 7.7%.

For a card of a given suit, it’s 13/52 or ¼, making it a 25% chance.

The important thing to remember is that, as cards are dealt, the odds will alter and you need to subtract from your calculation any known cards that have been played.. This is easiest explained with a single deck game like poker.

Step 3: Understanding poker outs

In poker, the outs are the cards which you need dealt in order for your hand to become the strongest possible. For instance, if you hold two spades and there are two more on the turn then you only need one more to come out in order to have a flush. Even if you already have three of a kind or a pair, a flush is stronger and more likely to win. The out in this case is any card of the same suit.

You might assume the odds are 25% but with 6 cards already out, you need to subtract that from the percentage. You know 6 cards have been dealt, so there are 46 remaining and 4 of them are of the suit you need, meaning there are 9 suited cards left.

9/46 gives you the percentage chance you have of receiving the card you need, which is approximately 19%.

When it comes to calculating this, you only count the cards you’ve seen, so the other players’ hands aren’t taken into consideration (because you can’t know what they are).

Step 4: Quick calculations

The rule of four and two is a useful method to approximate the odds whilst playing. You won’t get the exact probability but you get a close guess while you’re deep in a game. This is how it works: when you’ve only seen the flop, you multiply the out by 4 and if you’ve just seen the turn you multiply by 2.

Looking at the results you get, you might be curious why our previous estimate of 19% has been replaced by 36% for the first draw and 18% for the second. As you have two chances of the card you draw being the one you need, the probability is doubled for the first draw but with just one card left to come out, it then reverts to 18% for the next.

While it isn’t exact, it gives you a good approximation.

Step 5: Other games

Now, when it comes to other casino games it becomes a little harder to work out the odds. Games such as blackjack and baccarat use multiple decks in order to provide suitably random results and, in blackjack’s case, prevent card counting. You can still use the rough estimate of 1/52 for a certain card but with multiple decks, the exact number can change dramatically. Keeping a rough idea of the odds still helps here, it’s the same mechanism used in blackjack basic strategy and having it in mind can help you decide on the best play.

Do you think you’re ready to put this into practice? Why not play a few hands and see how you do!