How many different games of solitaire are possible? And what is the chance of winning any one of them?
Marshall Brain Answers:
In case you have never seen solitaire before, here is a game being played:
A well shuffled deck of cards has a lot of possibilities. How many exactly? You can think of it this way. The first card you put down can be any one of 52 possible cards. The next card can be any one of the 51 remaining cards. The next can be any one of the remaining 50, and so on. This is expressed mathematically as 52! (52 factorial) possibilities for different arrangements of the cards in a shuffled deck. That means:
52! = 52 * 51 * 50 * ... * 3 * 2 * 1 = 8.07 * 10^67
That's 8 with 67 zeros after it. Which is a lot of possible arrangements for the cards. Think about it this way. Let's say you took 7 billion human beings (roughly the number of humans on the planet today) and you had them play 100 games of solitaire each, every single day, for the next 10,000 years. In other words, the entire human race will be doing nothing but playing solitaire for 10,000 years. How many games is that? It's:
7,000,000,000 * 100 * 365 * 10,000 = 2,555, 000,000,000,000,000 = 2.555 * 10^18
In other words, all of that solitaire playing does not even begin to put a dent in all the possible games of solitaire.
In other words, assuming well-shuffled decks of cards, it is quite possible that every game of solitaire ever played has been unique. If you shuffle a deck of cards, lay it out and play a game of solitaire right now, chances are that you are the only person who has ever played that particular configuration of cards, and it will never be played again.
So what are your chances of winning that game? The only way to answer that question is to play lots of games of solitaire and see what the average is. Fortunately someone has taken the time to do that, and here are his results:
He claims that if you are playing normal "Draw 3 cards, Keep going around deck" solitaire (as shown in the video above), your odds of winning are roughly 1 out of 6 games.