When Wild Bill Hickok was killed in a Deadwood bar, he was playing poker. He was holding two aces, two eights and a queen----now known, unsurprisingly, as "Dead Man's Hand".

But the chance of having that particular hand (assuming that all hands have equal probability) is:

ACE + ACE + EIGHT + EIGHT + QUEEN

4/52 * 3/51 * 4/50 * 3/49 * 4/48 * (13C4) = 0.0023

So 0.2% of all hands will be a Dead Man's Hand. So if he played at least 1000 games of poker, he would have obtained a Dead Man's Hand at least twice, and so the "unsurprisingly" loses some of its irony.

I don't think so. I can see where you're coming from, but the (13C4)

term doesn't make sense to me. The bit before that is the probability

of getting the 5 cards dealt in the order AA88Q, and it looks like the

(13C4) is an attempt to adjust for the different possible orders, but

it doesn't make sense.

What you need to do for the last term is to multiply the probabilities

by the number of different orders you could get AA88Q in. It doesn't

matter which suits they are, as you've already allowed for that in the

earlier calculation (4/52 refers to any ace).

The actual number of ways you can arrange AA88Q is 30. This comes from

the number of ways to arrange 5 cards = 5! = 5*4*3*2*1=120. Then since

you have 2 aces you have to divide this by 2! (which equals 2), to get

60, and then by 2! again for the eights to get 30.

So the actual probability is 4/52 * 3/51 * 4/50 * 3/49 * 4/48 * 30 =

0.0000554 , or approximately a one in eighteen thousand chance.

We can confirm this by thinking about it slightly differently. We can

calculate how many different poker hands there are, i.e. the number of

ways to choose 5 cards from 52. This is the calculation (52 C 5) which

can be calculated as 2598960. Then we need to work out how many of

these are the dead man's hand. There are 6 possible ways to choose two

aces, 6 possible ways to choose two eights, and 4 possible queens.

Multiply these together 6x6x4 = 144 possible dead man's hands.

So the probability of a dead man's hand is 144/2598960 = 0.0000554,

the same as above.