Wizard in error?

This problem and solution are posted on the Ask the Wizard Texas Hold'em probability page, and I cannot see how it can possibly be right. Am I missing something very important, or has the wizard got this one totally wrong?


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Playing Texas Hold’em with 10 players using a standard 52 card deck, after the first two cards are dealt to each player, what are the odds that the "flop" - the next three cards - will all be the same suit? Does it make a difference if my hand has both cards of the same suit and/or each one a different suit?
— Mark from Milford

Before considering your own cards the probability is 2*(2/17) = 23.529%, as I just explained in an earlier question. If you have one of each color the probability that the flop is all the same color is 2*combin(25,3)/combin(50,3) = 2*(2300/19600) = 23.469%. If you have two of the same color the probability is (combin(26,3)+combin(24,3))/combin(50,3) = (2600+2024)/19600 = 23.592%.

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23%?? Surely that's way too high.

My answer to the first part of the question is (12/51)*(11/50)~4%.

It looks to me that the answer provided shows the calculations for getting two of the same color, that is a group of hearts/diamonds or a group of clubs/spades.

But, this doesn't sound like what the question was asking.

If you don't consider your hand, the probability of getting 3 cards of the same suit are 4*combin(13,3)/combin(52,3) = 0.051764706 (or 22/425) .

Thanks. Just wanted to report that. Probably best for the answer to be edited.

Our answers are equivalent (12/51)*(11/50)=4*(13c3)/(52c3)

The math totally escapes me.

But, having played my fair share of Hold 'Em, I'd say the odds of a suited flop are WAY lower than 23%. I wouldn't doubt that it's around 4%.

Note that the Wiz uses the term "Color" in his answer, not "Suit".

Personally, I think 23% seems about right for a flop that is all red or black.


Bottom line: I think there is an error. But the response may be correct for a "colored" flop....

That error was due to the wrong answer being associated to that question. The site recently went through a complete overhaul, and that question was evidently a casualty. I just fixed it.

Also, (12/51)*(11/50) =~ 5.18%.

If you have two cards of the same suit, the probability is (3*COMBIN(13,3)+COMBIN(11,3))/COMBIN(50,3) = 0.052193878 .

If you have two different suits, the probability is (2*COMBIN(13,3)+2*COMBIN(12,3))/COMBIN(50,3) = 0.051632653 .

Quote: kerzane

This problem and solution are posted on the Ask the Wizard Texas Hold'em probability page, and I cannot see how it can possibly be right. Am I missing something very important, or has the wizard got this one totally wrong?

https://wizardofodds.com/ask-the-wizard/texas-hold-em/probability/

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Playing Texas Hold’em with 10 players using a standard 52 card deck, after the first two cards are dealt to each player, what are the odds that the "flop" - the next three cards - will all be the same suit? Does it make a difference if my hand has both cards of the same suit and/or each one a different suit?
— Mark from Milford

Before considering your own cards the probability is 4*(2/17) = 23.529%, as I just explained in an earlier question. If you have one of each color the probability that the flop is all the same color is 2*combin(25,3)/combin(50,3) = 2*(2300/19600) = 23.469%. If you have two of the same color the probability is (combin(26,3)+combin(24,3))/combin(50,3) = (2600+2024)/19600 = 23.592%.

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23%?? Surely that's way too high.

My answer to the first part of the question is (12/51)*(11/50)~4%.

The Wizard did the math for the same color instead of suit.
Before considering your own cards the probability is 4*combin(13,3)/combin(52,3) or about 5.1764706 % (12/51)*(11/50) is correct.
If your cards are the same suit its:( 3*combin(13,3)+combin(11,3))/combin(50,2) or about 5.2193878 %
If your cards are 2 different suits its:( 2*combin(13,3)+2*combin(12,3))/combin(50,2) or about 5.1632653 %

Edit: I type slow :)

Quote: Wizard

That error was due to the wrong answer being associated to that question. The site recently went through a complete overhaul, and that question was evidently a casualty. I just fixed it.

Ok thanks.

Quote: Wizard

Also, (12/51)*(11/50) =~ 5.18%.

Yeah, that was a very quick guesstimate, I just meant to point out that it was far from 23%.

Cheers.