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How to calculate your chance of winning: Bridge example

Bridge example

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1. The game we play is Bridge; we want to get a Royal Flush (that is 13 spades) Such a hand is perhaps reported two dozen times a year in the U.S. I.e. it doesn't happen often!
2. There are 52 cards in a deck, of which 13 are Spades.

Keep in mind: A very good hand is just as likely as getting a bad hand, so the chance of getting 13 spades is the same as getting any other group of thirteen cards, spades or not.

Since there are 13 to a hand, and 52 cards to a deck, the question we are asking is how many different 13-combinations are possible within 52 things.

The 1st card dealt to you may be any of the 13 spades without preventing you from getting the prize hand we are talking about. So your chance of starting out right is 13/52, which is P = 0.25.
Of the 51 remaining cards only 12 are spades. So your chance with the second card is 12/51. With the next card your chance is 11/50, and so on... till you get your 13th card.
Multiply together these fractions according to the rule and you will find that the chance of getting a Royal Flush is 1 in 653,013,559,600 hands. Not a big chance...

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