# odds of pair in 3card?

#### 3mptyseat

##### Quit Gambling
PABnonaccred
I guess I wasn't sure where to post this.
But I figured, "CM is hecka smart... and so he'd probably know... But if not, a helpful Meister math person would surely step up..."

So my question is this, playing three card poker, what are the odds of having a pair in any given hand? I know that, much like the odds in a truly random slot are static in one spin as they are in the very next, and the same as in 500 spins later, the likelihood that any event happening back to back to back, for example, can be expressed in a ratio...

What is the likely hood of one player having 5 consequtive hands with a pair? And how 'bout 6 of 7 hands with a pair?

My math for the first question is something like this I think: Dealt 1 card, on the second we have 3/51 chance of pairing and on the third card, 2 chances of 3/50. So for simplicity, 3/50 + 3/50 + 3/50 = ~9/50 or 20% chance of making a pair in 3 cards....

So I think that the likelihood of 5 consecutive hands pairing would be 1/5 x 1/5 x 1/5 x 1/5 x 1/5= .032 % chance of happening? So a lil over once in 3000 times? Is that right? I have no idea how to figure out 6 of 7, and I know the answer is static on a hand to hand basis...

#### baldidiot

##### Full Member
webmeister
I guess I wasn't sure where to post this.
But I figured, "CM is hecka smart... and so he'd probably know... But if not, a helpful Meister math person would surely step up..."

So my question is this, playing three card poker, what are the odds of having a pair in any given hand? I know that, much like the odds in a truly random slot are static in one spin as they are in the very next, and the same as in 500 spins later, the likelihood that any event happening back to back to back, for example, can be expressed in a ratio...

What is the likely hood of one player having 5 consequtive hands with a pair? And how 'bout 6 of 7 hands with a pair?

My math for the first question is something like this I think: Dealt 1 card, on the second we have 3/51 chance of pairing and on the third card, 2 chances of 3/50. So for simplicity, 3/50 + 3/50 + 3/50 = ~9/50 or 20% chance of making a pair in 3 cards....

So I think that the likelihood of 5 consecutive hands pairing would be 1/5 x 1/5 x 1/5 x 1/5 x 1/5= .032 % chance of happening? So a lil over once in 3000 times? Is that right? I have no idea how to figure out 6 of 7, and I know the answer is static on a hand to hand basis...

Quick question - do you mean the odds of getting exactly one pair, or the odds of getting at least one pair? (ie: winning the bonus bet on 3 card poker)

These are quite different things but I'm assuming the chance of a win is more important to you? Plus you said the chance of it containing a pair, so technically 3 of a kind contains a pair... What about straight/flush?

I'll split the answer into parts over a couple of posts to make it easier to read...

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#### baldidiot

##### Full Member
webmeister
Taking it literally, the chance of getting EXACTLY one pair (eg: three of a kind doesn't count) are:

[chance 1st card matches x chance 2nd card DOESN'T match] + [chance 1st card DOESN'T match x chance 2nd card matches] + [chance that 3rd card matches 2nd card]

which is:

[3/51 x 48/50] + [48/51 x 3/50] + [48/51 x 3/50]

= approx 1 in 5.9

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#### baldidiot

##### Full Member
webmeister
The chance of getting at least one pair, but excluding straights and flushes (eg: pair or 3oak) is:

[chance 1st card matches x chance 2nd card DOESN'T match] + [chance 1st card DOESN'T match x chance 2nd card matches] + [chance 1st card matches x chance 2nd card matches] + [chance that 3rd card matches 2nd card]

which is:

[3/51 x 48/50] + [48/51 x 3/50] + [3/51 x 2/50] + [48/51 x 3/50]

= approx 1 in 5.82

#### baldidiot

##### Full Member
webmeister
Next, chance of getting 5 consecutive hands in a row with a pair:

Exactly one pair:
5.9 ^ 5 = 1 in 7166

or

Contains a pair (inc 3oak):
5.82 ^ 5 = 1 in 6689

#### baldidiot

##### Full Member
webmeister
Finally, the chance of getting 6 out of 7 can be calculated like this:

[chance win ^ 6] x [1 - chance win] / [possible combinations - because order does not matter]

which is either:

exactly one pair:
(1/5.9 ^ 6 x 1-1/5.9) / 7 = approx 1 in 7275

or

contains a pair (inc 3oak):
(01/5.82 ^ 6 x 1-1/5.82) / 7 = approx 1 in 6717

Let me know if you want the odds of any win on the pair plus (which is what I'm assuming you're getting at). Ie: including straights and flushes

#### 3mptyseat

##### Quit Gambling
PABnonaccred
I had a situation where I was playing for awhile at one bet with mixed results and the I raised the bet, and then I got ak or one pair 5 times in a row. The dealer had a better pair each of those 5 hands. We then pushed. On the 7th hand at the higher stake, I then drew a K high, and they had another pair. Given that I was qualified in those 7 hands, it even less likely, so you have answered my question. And while the odds are by no means impossible, its fairly unlikely that I was in a random environ. What you lack in hair, you make up for with a lack of idiocy. TYVM BI.

#### 3mptyseat

##### Quit Gambling
PABnonaccred
@baldidiot

what i can say to that .. if you think at the same its about 100 player playing on 3 card poker ... that would make this outcome every 700 deals...

This is what the csr said when I asked how he felt about the hands I enjoyed last night. I explained the 1:7166 over 5 str8 hands, and that was his respuesta?

I can't discount that with my limited brain power... But is that right? It seems like the amount of deals would be if you counted all 100 ppls deals as one deal... Which means that he just blew smoke str8 up my skirt. But, I will admit, it did sooth me a lil bit...

#### baldidiot

##### Full Member
webmeister
I have no idea what the csr is going on about, sounds like they just don't know what to say.

A 1 in ~7,000 event happening isn't that rare - sucks that it happened when you raised your bet, but it happens.

I'm assuming you're playing at a legit site with legit software, such as one of the CM accredited casinos. If so then there's nothing to be concerned about.

#### 3mptyseat

##### Quit Gambling
PABnonaccred
Heartwarming theory bi but anything goes with us facing sites ATM.

#### baldidiot

##### Full Member
webmeister
Heartwarming theory bi but anything goes with us facing sites ATM.

Oh

#### GrandMaster

##### Ueber Meister
CAG
So my question is this, playing three card poker, what are the odds of having a pair in any given hand?
You could do worse than look at