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Solitaire (Card Game) Question

Posted: Mon Feb 24, 2014 8:05 am
by rogerwimmer
Doc: I like to play solitaire on my computer and phone and every so often I win a game very quickly – probably in about a minute or so. When that happens, I wonder what the chances are of getting that same hand dealt to me. Any ideas? - JD

JD: The simple answer to your question is: Not very likely. Here is a longer answer . . .

To compute the odds of dealing the same hand in solitaire or any other card game that uses a full deck of 52 cards, you need to use a mathematical operation called a factorial, which is designated as n!, where “n” is a number (integer) followed by an exclamation point. Factorials are used in many math and statistics operations, but one use of factorial is to provide the number of possible displays of a data set, or group/list of items.

For example, let’s say you have three playing cards – Ace, 2, and 3. How many different ways can those three cards be displayed (dealt)? The formula for 3! (“three factorial”) is 3 x 2 x 1, or 6. Here are the six ways the three cards can be dealt:

A 2 3
A 3 2
2 A 3
2 3 A
3 A 2
3 2 A

Got it? A factorial is the product of all positive integers less than or equal to n. So, if you had five playing cards, the number of different displays, or ways the cards could be dealt is 5! (“five factorial”), which is 5 x 4 x 3 x 2 x 1, or 120.

The factorials for 3 and 5 are relatively small, but now let’s consider a full deck of 52 cards. The formula for 52 factorial is 52 x 51 x 50 x 49 . . . etc. . . . x 1, and that product is (No, I’m not kidding):

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000

That number is: 80 unvigintillion, 658 vigintillion, 175 novemdecillion, 170 octodecillion, 943 septendecillion, 878 sexdecillion, 571 quindecillion, 660 quattuordecillion, 636 tredecillion, 856 duodecillion, 403 undecillion, 766 decillion, 975 nonillion, 289 octillion, 505 septillion, 440 sextillion, 883 quintillion, 277 quadrillion, 824 trillion.

So, your chance of dealing the same hand in solitaire (or any other card game) is ONE in that huge number . . . one in 80.658 unvigintillion. Not impossible, but it’s highly unlikely that you’ll ever see the same hand again. In fact, playing cards were invented in China during the 9th century, and all the hands ever dealt for every card game ever played by the world’s population would not approach this number…and never will.

It’s easy to find a factorial on the Internet. Just go to a search engine and type in, for example, 52! or 52 factorial. However, the problem is that you’ll get the number in scientific notation. Google and Bing show 52! (“52 factorial”) as 8.0658175e+67. That’s boring. To get all the numbers, you can go to one of several factorial calculators. Some of them require that you select the “Full Integer” option to see all the numbers.

However, the best option is to use Wolfram Alpha. Look at the information provided when you search for 52! – click here.

In summary, go ahead and deal away because it’s very unlikely that you’ll see the exact same hand twice.


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