Betting odds

You are offered a wager. I'll shuffle an ordinary deck and deal you 13 cards. If none of your cards ranks above 9, he'll give you a thousand pounds. Otherwise you must give him one pound.

Is this a good bet?

Lord Yarborough gave his name to a hand of cards dealt in bridge that has no card higher than a nine. The probability of getting a Yarborough is about 1828 to 1. The Earl offered 1000 to anyone who achieved a "Yarborough" on condition they paid him 1 each time they did not succeed.

Not only the odds against a particular player getting a Yarborough about 1828.04 to 1, but the odds against some two players getting Yarboroughs are about 91 million to 1.

To compute the odds that a particular player gets a Yarborough, note that 32 of the 52 cards are nine or less. The chance that the first card is nine or less is 32/52; given that, the chance that the second card is nine or less is 31/51; and so on. The chance that all 13 cards are nine or less is

32/52 * 31/51 * 30/50 * 29/49 * 28/48 * 27/47 * 26/46 * 25/45 * 24/44 * 23/43 * 22/42 * 21/41 * 20/40


or about 1/1828, for odds of about 1827 to 1. The chance that two particular players both have Yarboroughs, that their 26 cards are nine or less, is

32/52 * 31/51 * 30/50 * 29/49 * 28/48 * 27/47 * 26/46 * 25/45 * 24/44 * 23/43 * 22/42 * 21/41 * 20/40 in turn multiplied by

19/39 * 18/38 * 17/37 * 16/36 * 15/35 * 14/34 * 13/33 * 12/32 * 31/11 * 30/10 * 29/9 * 28/8 * 27/7

which works out at roughly 91 million to one.