Poker and the Laws of Probability

During a poker game have you ever wondered why a five-card hand that contains five cards from the same suit is better than one that has two sets of matching ones? What is the determining factor in the ranking of various poker hands? The answer is probability. In the world of five-card stud the value of a particular set of cards is based solely on the likelihood of a player obtaining them.

Beginning with the basic rules

Unfortunately there is a significant learning curve required before the probability of five-card hands can be determined. An excellent starting point for developing these skills would be to calculate probabilities when selecting a single card from a standard deck. Probability is defined as the ratio of possible successes over the total possibilities (sum of possible successes plus possible failures). The standard deck contains the following 52 cards:

Clubs (black): ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king

Spades (black): ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king

Hearts (red): ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king

Diamonds (red): ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king

There are four suits each containing 13 cards.

A few warm up exercises

When drawing one card what would be the probability of it being the six of diamonds? There is only one six of diamonds and any single selection would have 52 possible outcomes. Thus the possible successes would be one and the total possibilities would be 52. The probability of drawing the four of diamonds would be 1/52 which is 0.019 or 1.9%.

What would be the probability of picking a six? The possible successes have now increased to four. Thus the ratio would be 4/52 which would be 0.076 or 7.6%. (Note that this answer is exactly four times the 1.9% from the previous problem.

What would be the probability of picking a diamond? With 13 possible successes the probability would jump to 25% (13/52).

Ratcheting up the difficulty

Increasing the number of cards selected to two or more results in significant mathematical complications revolving around the actual order of the cards. In probability there are two different ways to measure the number of possible outcomes — permutations and combinations. A permutation is a group of elements with no repetitions that require a specific order. Arranging books in alphabetical order on a shelf is a permutation since each book is unique and the order is specific. A combination is also a group of non-repeating elements but their order is disregarded. A poker hand is a combination since the precise order that the cards are received is not a consideration in determining its overall value. The card game “War” would be a permutation since the order is critical.

Another mathematical concept that will be utilized is the “factorial”. In math factorial is symbolized by an “!” and indicates multiplication of all whole numbers in descending order. For example 4! equals 4 x 3 x 2 x 1.

Permutations versus combinations

When choosing two numbers from the array of 22, 23, and 24 there are six possibilities:

22, 23

22, 24

23, 24

23, 22

24, 22

24, 23

If order is a consideration these are all unique creating six possible permutations. However, when order is disregarded the first three contain the same numbers as the last three. Thus, there would be half as many combinations. By repeating this process, choosing three numbers out of four, then four out of five and so on, a pattern will emerge. The number of combinations is equal to the number of permutations divided by the factorial of the number of terms being utilized. In the first example that would be 6 divided by 2! which gives the result of 3.

Choosing two cards

Since order will not matter in the following problems, all will be considered combinations.

What is the probability of picking two cards and neither match? The potential successes would begin with 52 options for the first selection. The three remaining matching cards cannot be chosen leaving 48 for a successful second choice. The possible successes would be 52 multiplied by 48 divided by 2! which equals 1248. This represents the potential successes (numerator). The total number of potential successes and failures (the denominator) would be 52 (options for first card) multiplied by 51 (options for second card) divided by 2! resulting in 1325. Consequently, the probability of drawing two non-matching cards would be 1248/1325 which is 0.941 or 94.1% – a highly likely outcome.

What is the probability of selecting two hearts? The first card would have 13 potential hearts to select; the second would be reduced to 12. The possible successes would be 13 times 12 (156) divided by 2! (78). Dividing 78 / 1325 (the denominator will remain the same for every two-card pairing) results in 0.059 or 5.9% — significantly less likely.

The lightning round

Find the probability of selecting two cards of the same suit. The first option is 52 but the second would be limited to the12 remaining cards of that card’s suit. Multiplying 52 and 12 then reducing by 2! gives 312 which divided by 1325 is 0.235 or 23.5%.

What is probability of selecting two sixes? The first option is four (total sixes in a deck) times the three remaining sixes divided by 2! equals six. After dividing by 1325 the probability is 0.45%. For any two matching cards the first option would be 52, the second three (remaining matches) and after the factorial the numerator would be 78 and the probability would be (5.9%).

A homework assignment

Based on this information what would be the probability of a five-card poker hand no matches? How about the chances of two cards matching? (The rounded answers are 50.7% and 42.3%.)