Hack 42. Play with Cards and Get Lucky

While it is true that Uncle Frank spends much of his time in taverns using dice to win silly bar bets and smiling real charming-like at the ladies, there is more to his life than that. For instance, sometimes he uses playing cards instead of dice.

People, especially card players, and especially poker players, feel pretty good about their level of understanding of the likelihood that different combination of cards will appear. Their experience has taught them the relative rarity of pairs, three-of-a-kind, flushes, and so on. Generalizing that intuitive knowledge to playing-card questions outside of game situations is difficult, however.

My stats-savvy uncle, Uncle Frank, knows this. Sometimes, Uncle Frank uses his knowledge of statistics for evil, not good, I am sorry to say, and he has perfected a group of bar bets using decks of playing cards, which he claims helped pay his way through graduate school. I'll share them with you only for the purpose of demonstrating certain basic statistical principles. I trust that you will use your newfound knowledge to entertain others, fight crime, or win inexpensive nonalcoholic beverages.

Getting a Li'l Flush

In poker, a flush is five cards, all of the same suit. For my Uncle Frank, though, there is seldom time to deal out complete poker hands before he is asked to leave whatever establishment he is in. Consequently, Uncle Frank often makes wagers based on what he calls li'l flushes.

The bet

A little flush (oops, sorry; I mean li'l flush) is any two cards of the same suit. Frank has a wager that he almost always wins that has to do with finding two cards of the same suit in your hand. Again, because of time constraints, his poker hands have only four cards, not five.

The wager is that you deal me four cards out of a random deck, and I will get at least two cards of the same suit. While this might not seem too likely, it is actually much less likely that there would be four cards of all different suits. I figure the chance of getting four different suits in a four-card hand is about 11 percent. So, the likelihood of getting a li'l flush is about 89 percent!

Why it works

There are a variety of ways to calculate playing-card hand probabilities. For this bar bet, I use a method that counts the number of possible winning hand combinations and compares it to the total number of hand combinations. This is the method used in "Play with Dice and Get Lucky" [Hack #43].

To think about how often four cards would represent four different suits, with no two-card flushes amongst them, count the number of possible four-card hands. Imagine any first card (52 possibilities), imagine that card combined with any remaining second card (52x51), add a third card (52x51x50) and a fourth card (52x51x50x49), and you'll get a total of 6,497,400 different four-card hands.

Next, imagine the first two cards of a four-card hand. These will match only .2352 of the time (12 cards of the same suit remain out of a 51-card deck). So, about one-and-a-half million four-card deals will find a flush in the first two cards. They won't match another .7648 of the time. This leaves 4,968,601 hands with two differently suited first two cards.

Of that number of hands, how many will not receive a third card that does not suit up with either of the first two cards? There are 50 cards remaining, and 26 of those have suits that have not appeared yet. So, 26/50 (52 percent) of the time, the third card would not match either suit.

That leaves 2,583,673 hands that have three first cards that are all unsuited. Now, of that number, how many will now draw a fourth card that is the fourth unrepresented suit? There are 13 out of 49 cards remaining that represent that final fourth suit. 26.53 percent of the remaining hands will have that suit as the fourth card, which computes to 685,464 four-card combinations with four different suits. 685,464 divided by the total number of possible hands is .1055 (685,464/6497400).

There's your 11 percent chance of having four different suits in a four-card hand. Whew! By the way, some super-genius-type could get the same proportion by using just the relevant proportions, which we used along the way during our different counting steps, and not have to count at all:

Finding a Match with Two Decks of Cards

You have a deck of cards. I have a deck of cards. They are both shuffled (or, perhaps, souffl\x8e d, as my spell check suggested I meant to say). If we dealt them out one at a time and went through both decks one time, would they ever match? I mean, would they ever match exactly, with the exact same cardfor example, us both turning up the Jack of Clubs at the same time?

The bet

Most people would say no, or at least that it would certainly happen occasionally, but not too frequently. Astoundingly, not only will you often find at least one match when you pass through a pair of decks, but it would be out of the ordinary not to. If you make this wager or conduct this experiment many times, you will get at least one match on most occasions. In fact, you will not find a match only 36.4 percent of the time!

Why it works

Here's how to think about this problem statistically. Because the decks are shuffled, one can assume that any two cards that are flipped up represent a random sample from a theoretical population of cards (the deck). The probability of a match for any given sample pair of cards can be calculated. Because you are sampling 52 times, the chance of getting a match somewhere in those attempts increases as you sample more and more pairs of cards. It is just like getting a 7 on a pair of dice: on any given roll, it is unlikely, but across many rolls, it becomes more likely.

To calculate the probability of hitting the outcome one wishes across a series of outcomes, the math is actually easier if one calculates the chances of not getting the outcome and multiplying across attempts. For any given card, there is a 1 out of 52 chance that the card in the other deck is an exact match. The chances of that not happening are 51 out of 52, or .9808.

You are trying to make a match more than once, though; you are trying 52 times. The probability of not getting a match across 52 attempts, then, is .9808 multiplied by itself 52 times. For you math types, that's .9808⁵².

Wait a second and I'll calculate that in my head (.9808 times .9808 times .9808 and so on for 52 times is...about...0.3643). OK, so the chance that it won't happen is .3643. To get the chance that it will happen, we subtract that number from 1 and get .6357.

You'll find at least one match between two decks about two-thirds of the time! Remarkable. Go forth and win that free lemonade.