# Math help: Chance to hit a pair on flop?

• Bronze
Joined: 20.11.2012
I was replying hands in PO and noted that it says that you get 40.408% chance to hit a pair on flop (ie AK as your cards).

I'm still a newbie on poker math but remember that I got it to around 36% when calculating myself...

Shouldn't the chance be 6/50+6/49+6/48 = 36.745% to flop a pair?

(6 is the outs and 50/49/48 is the cards left in the deck)
• 7 replies
• Global
Joined: 23.11.2009
Yep, if you don't know anything about your opponents cards or dead cards then unpaired holecards have a 6/50 + 6/49 + 6/48 = 36.7% chance of flopping a pair.

Do the replayed hands factor out known holecards? That may be where the confusion lies.

edit: my math is wrong. JB is right.
• Bronze
Joined: 22.11.2009
It's 1-(chance you miss all three cards) = 1 - (44/50)*(43/49)*(42/48) = 32.4%
• Bronze
Joined: 12.10.2011
It's 1-(chance you miss all three cards) = 1 - (44/50)*(43/49)*(42/48) = 32.4%
This pretty much.

But if you want to know the probability of hitting exactly one pair (the above formula disregards the fact that you can flop more than 1 A or K), what you really have to look at is the amount of flop combinations that give you only one A or one K (duh). There is 50 * 49 * 48 / 3 * 2 * 1 = 19600 flop combinations. Then there's six out of 50 cards that give you a pair, meaning the other 2 cards can not be an A or K. There's 44 * 43 / 2 * 1 = 946 such 2-card combinations. Since there's 6 aces and kings left (three of both), that's 946 * 6 = 5676 flop combinations that give you exactly one pair.

5676 / 19600 = ~28.95% chance to hit exactly one pair on the flop.

But the other formula also includes hitting things like two pair, trips, etc.
• Global
Joined: 23.11.2009
oh gosh, ty jb. No idea what i was thinking. time to grind probability maths again.
• Bronze
Joined: 27.09.2009
Thanks all, moved to beginners' questions
• Bronze
Joined: 26.12.2009
Originally posted by TinoLaan
It's 1-(chance you miss all three cards) = 1 - (44/50)*(43/49)*(42/48) = 32.4%
This pretty much.

But if you want to know the probability of hitting exactly one pair (the above formula disregards the fact that you can flop more than 1 A or K), what you really have to look at is the amount of flop combinations that give you only one A or one K (duh). There is 50 * 49 * 48 / 3 * 2 * 1 = 19600 flop combinations. Then there's six out of 50 cards that give you a pair, meaning the other 2 cards can not be an A or K. There's 44 * 43 / 2 * 1 = 946 such 2-card combinations. Since there's 6 aces and kings left (three of both), that's 946 * 6 = 5676 flop combinations that give you exactly one pair.

5676 / 19600 = ~28.95% chance to hit exactly one pair on the flop.

But the other formula also includes hitting things like two pair, trips, etc.
Sick! Thanks Tino you just destroyed my brain
• Bronze
Joined: 12.10.2011
Originally posted by RasTweet
Sick! Thanks Tino you just destroyed my brain
You're welcome