# Odd for drawing an 8 in 5 card stud

At my weekly poker game two friends argued about the odds of drawing an 8 in 5 card stud, where 8s were wild.

It seems like after research it would be about 34% but wanted to get! a second opinion.

There were 6 players at the table, so 30 cards were drawn, 5 to each player. I calculated that there is a 97.3 percent chance an 8 is drawn when sample of 30 cards are taken from a 52 card deck. One person said each player has equal chance 97.3 / 6 is 16.2 but that's not right either in my mind.

Let `C(n,m) = n!/(n - m)!`. That is, the number of ways to choose `m` items from `n`.

The calculation you did for the first value is correct:

``````1 - C(52-4, 30)/C(52, 30) = 97.3%
``````

You can arrive at that calculation by first calculating the probability that no eights are drawn. That is the number of ways to choose 30 cards from the deck excluding the 4 eights, divided by the total number of ways to choose 30 cards from the deck. Then since you want the opposite of that, subtract it from 1.

The probably of getting an eight for any one player is 34.1%. You can calculate it the exact same way:

``````    1 - C(52-4, 5)/C(52, 5) = 34.1%
``````

One person said each player has equal chance 97.3 / 6 is 16.2

Probabilities don't work like that. For example, if you flip a coin 6 times you have a `98.4375%` that you'll get heads at least once. That doesn't mean that the probability of getting heads on a single flip is `98.4375/6 = 16.4%`. The probability is 50%.