Starting hand probabilities

[MathBabe]

We all know there are 169 starting hands in Hold'Em. But the odds of getting each hand are NOT 168:1. I've been meaning to write this up for a while, and I think I've worked it out in a way that isn't too confusing.

(Executive summary: odds of getting a specific pocket pair are 220:1, two specific suited cards 330:1, and two specific unsuited cards 110:1. Nice easy numbers to remember).

The number of starting hands is easily calculated by 52*51/2 (52 possible cards for the first card, times 51 possible cards for the second, divided by 2 since we don't care which order we get them in). That's 1326. BUT, in practice we don't care if we have AsAc or AsAh - they're both AA. That's what cuts the number down, and understanding how gives us the probability of each type of hand.

For each pocket pair, there are 6 versions you could have based on suits. That's odds of 1320:6, or 220:1 (a probability of .45%). The odds of getting dealt a pocket pair on any hand are 13 times that, 1238:78, or 16:1 (5.9%).

For each two suited cards, like JTs, there are only 4 versions you could have (one for each suit). So, the odds are 1322:4 or 331:1 (.3%). There are 13*12/2 combinations of two unpaired cards, which is 78, so the total odds of getting any two suited cards are 1014:312 or about 3:1 (23.5%).

For each two unsuited cards, like AKo, there are 12 versions you could have (4 possible suits for the first card, times 3 for the second - we don't divide by 2 because we DO care about order in this case). So, for each particular unsuited cards, you have odds of 1314:12 or 110:1 (.9%). There are 13*12/2 =78 combinations of suited cards, so the total odds are 300:936 or about 1:3 FOR it happening. (70.6%).

We can check by adding them all up. 78 pocket pairs + 312 suited cards + 936 unsuited cards = 1326, which matches the original number. Adding up the probabilities, we get 5.9% + 23.5% + 70.6% = 100%.

[MAYHEM45]

Great post MB. The 23.5% chance for any two suited really helps me see how these donkeys have VPIPs over 40.

[Talking Poker]

Sweet post, MB...

I'm not trying to nitpick here, but odds are usually expressed x:1 instead of 1:x. For example, you were a "3 to 1" underdog (the ":" = "to") not a "1 to 3" underdog. So in the future, or if you feel like editing this post, I think it would be a little more clear if you wrote them in the reverse order.

I know it sounds stupid, but I was reading your post very closely and kept getting messed up because of that.

[MathBabe]

Nitpick away! It's encouraged!

I keep getting confused with this myself. Technically (which is to say, mathematically), which way you write it depends on what you're expressing. If there's a 70% chance of rain today, the odds of it raining are 7:3. The odds of it NOT raining are 3:7. So when I think of the "odds of getting a pair", I think if it as 1:16. The "odds against getting a pair" are 16:1.

I guess in poker, everything you want to happen is always unlikely. So the convention makes sense that you always assume you're the underdog, unless specifically stated otherwise. (Perhaps this is obvious if you follow sports betting, which I don't).

Looking in my poker books and surfing around a bit, it seems you're right, and it's no wonder I'm confused - it's ALWAYS expressed x:1, and you qualify by saying underdog or favourite. Now I see! (And I've edited to match).

[Talking Poker]

Exactly... the 1 goes on the right and then you qualify it, if necessary.

Edit: But your post doesn't look edited to me. I see all sorts of big numbers on the right still...