Odds question.

[OrianasDaad]

What are the odds of drawing the same holdem hand three times consecutively? Say Ah9h.

I get 2550^3, which is (51-1 * 50-1) ^3

Is this the correct way to do this?

[OrianasDaad]

I'm asking because it happened to me today, and I mentioned it at the table (I won the first and third hands with it.) Someone at the table said it "wasn't that uncommon" and "I should know why." (??)

Now, I know it isn't a common occurrence at all... but I could not be certain that my method to figure out how likely it actually is was correct. That's why I'm asking here.

[uuDevil]

The probability of being dealt Ah on the first card is (1/52). The probability of then being dealt 9h on the 2nd card is (1/51). Since we don't care about the order, the probability of Ah9h is (1/52)*(1/51)*2=0.000754148.

For 3 times in a row, this becomes .000754148^3. In terms of odds, (1-.000754148^3)/.000754148^3=2331473975 to 1.

As a check, there is 1 way to be dealt Ah9h, and C(52,2)=1326 ways to be dealt any 2 cards. 1/1326=0.000754148, same as above.

[DiceyPlay]

Your question could be ambiguous. You asked about getting the same hand (any hand) 3 times in a row and then you qualified it with Ah9h.

For Ah9h it would be

[p(you get Ah or 9h as first card) * p(you get the other card as second card)]^3

or [2/52 * 1/51]^3 ~ .0000000004 = Extremely rare.

But being dealt any two cards 3 times in a row is [2/52 * 1/51]^2 ~ .0000006 = a little less extremely rare.

The difference is you don't care what you were dealt on the first hand. You just want to know how likely it is to come up 2 more times.

But what happened to you only happens 4 in 10 billion for Ah9h specifically! Or 6 times in 10 million for any two cards.

[uuDevil]

[ QUOTE ]
I'm asking because it happened to me today, and I mentioned it at the table (I won the first and third hands with it.) Someone at the table said it "wasn't that uncommon" and "I should know why." (??)

Now, I know it isn't a common occurrence at all... but I could not be certain that my method to figure out how likely it actually is was correct. That's why I'm asking here.

[/ QUOTE ]If you were trying to do something like multiply odds, you can't do that directly. Calculate the probabilities first then convert to odds.

Unless you pick the hand ahead of time, maybe you should only be considering how likely you are to get it 2 times in a row.

As to whether you should know why, well some basic ability to calculate probabilities is no doubt a good thing. I wouldn't take anything said by a random poker player too seriously.