Odds of Flopping the Same Set 3 times in a Row

The craziest thing happened to me while playing a home game online last night. I was heads up with a buddy of mine and was severely short stacked(I had around 8K he had around 65K). I got dealt pocket 7s and went all in and he quickly called with Queens. I hit a set on the flop to stay alive. Next hand I get 7s again and go all in and he calls with Q8. He hits the queen on the flop but I hit a set of 7s on the flop again to win the hand. The very next hand I get pocket 7s again and raise, my buddy calls. I flop a set of 7s a third time! When we get to the river, I win against his AK(he hit a K on the flop to stay in the hand).

My question is: what are the odds of flopping a set with the same pocket pair 3 times in a row?

The wizard answered a previous question about what are the odds of getting the same pocket pair 3 times in a row(.000142108 or 1 in 7037):

/forum/gambling/poker/31884-odds-of-same-pocket-pair-three-times-in-a-row/

So would I take this number and multiply it by the chances of flopping a set 3 times(.12 * .12 * .12 = .001728)? When I multiply those numbers together I get .00000024192 or 1 in 4,133,598.

By the way, I did come back from an 8 to 1 deficit to win the game, mostly due to those 7s!

Quote: TheBobFather

The craziest thing happened to me while playing a home game online last night. I was heads up with a buddy of mine and was severely short stacked(I had around 8K he had around 65K). I got dealt pocket 7s and went all in and he quickly called with Queens. I hit a set on the flop to stay alive. Next hand I get 7s again and go all in and he calls with Q8. He hits the queen on the flop but I hit a set of 7s on the flop again to win the hand. The very next hand I get pocket 7s again and raise, my buddy calls. I flop a set of 7s a third time! When we get to the river, I win against his AK(he hit a K on the flop to stay in the hand).

My question is: what are the odds of flopping a set with the same pocket pair 3 times in a row?

The probability of getting a third card matching a pocket pair in the flop is 1 - (48/50 x 47/49 x 46/48) = 144 / 1225.
The probability of getting the same pocket pair in 3 consecutive hands is 1/17 (the probability of getting a pocket pair in the first hand) x 1/221 (the probability of getting the same pocket pair in the next hand) x 1/221 (the probability of getting the same pocket pair in the third hand) = 1 / 830,297.

OOPS - corrections are in red

The overall probability = (144 / 1225)^3 x 1 / 830,297 = about 1 / 511,000,000

Note that this applies to three specific hands, and not necessarily any three consecutive hands in a session.
Assuming you are dealt a pocket pair, the probability of getting a third card on the flop and then being dealt the same pair and getting a third card on the flop in both of the next two hands is this multiplied by 17, or about 1 / 30,000,000

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Quote: ThatDonGuy

The probability of getting a third card matching a pocket pair in the flop is 1 - (48/50 x 47/49 x 46/48) = 144 / 1225.
The probability of getting the same pocket pair in 3 consecutive hands is 1/17 (the probability of getting a pocket pair in the first hand) x 1/221 (the probability of getting the same pocket pair in the next hand) x 1/221 (the probability of getting the same pocket pair in the third hand) = 1 / 830,297.

The overall probability = 144 / 1225 x 1 / 830,297 = about 1 / 7,063,290.

Note that this applies to three specific hands, and not necessarily any three consecutive hands in a session.
Assuming you are dealt a pocket pair, the probability of getting a third card on the flop and then being dealt the same pair and getting a third card on the flop in both of the next two hands is this multiplied by 17, or about 1 / 415,488.

you need to cube the 144/1225 term, no?

Quote: unJon

you need to cube the 144/1225 term, no?

Why, yes - yes, I do. Corrections to the original post have been made.

Thanks for the detailed response! So for the purposes of bragging, I can say that this was a 1 / 511,000,000 chance of happening then? In other words, I had a better chance of winning Powerball and will probably never see it again?

Quote: TheBobFather

Thanks for the detailed response! So for the purposes of bragging, I can say that this was a 1 / 511,000,000 chance of happening then? In other words, I had a better chance of winning Powerball and will probably never see it again?

1 / 511,000,000 is for three specific hands. If you're talking three consecutive hands in general when you have been dealt a pair, it's 1/17 of that, or about 1 / 30,000,000.