Monotone flop...probability villain flopped a flush?

[Jouster777]

This seems like an elementary problem but there is controversy so I'd like the board to weigh in.

You get a flop of all spades. What is the probability that a single opponent started with 2 spades assuming you don't have 2 spades?

I say that the probability is 3.3% or 4.2% depending on whether you are looking at 0 or 1 spade in your hand. He says the probability villain has 2 spades is just under 6% no matter what the flop or your hand shows. Anyone want to help settle the point?

Here is my reasoning:
Against 1 player when you are looking at a monotone board you can account for 5 cards (yours and the board). There are either 9 or 10 cards unaccounted for in the board's suit (depending upon whether you hold 0 or 1 of that suit). The "pretest probability" (i.e. random cards with no other info applied) that villain holds 2 cards of that suit is:
When I have one of that suit = 9/47*8/46=3.3%
When I have 0 of that suit = 10/47*9/46=4.2%
(Cases where I have 2 of that suit is not the scenario presented)

I believe that his "just under 6%" number comes from not having information about any other cards in the deck (=13/52*12/51)

[LetYouDown]

Been a while since I cranked out numbers, so correct at leisure. Assuming your opponent holds a completely random hand.

You hold no spade:

C(10,2)/C(47,2) = 45/1081 = 4.16%

You hold one spade:

C(9,2)/C(47,2) = 36/1081 = 3.33%

[Jouster777]

Thanks...its always reassuring to see the same (almost) result using different but equally valid methods.

I guess the real question is even more basic. The joker I was arguing with was trying to say that the fact the flop was suited has no bearing on the likelihood that villain held suited cards - I assume because the likelihood of being dealt suited cards IN THE ABSENCE OF ANY OTHER INFORMATION is 5.9% and the hands are "dealt first".

Any comments on this line of thought?

[DrVanNostrin]

You're right about this. For example, let's say I hold A8 and I know my opponent holds either AA or KK. Before the flop KK is 2x as likely as AA (there are 6 ways for him to have AA and 12 to have KK). If the flop is A 7 2 he is 12x more likely to hold KK than AA (now there is only 1 way for him to have AA and still 12 to have KK).

[Jouster777]

Thanks...Its a pretty basic concept but this all this guy could say is how I needed an education.

I tried to explain it to him as: "imagine the dealer took 12 cards off the top of the deck (6 players, no burn), dealt the flop first, and then dealt the hands...is this any different than the situation we were dealing with?" [of course not]

Some people just can't handle logic that conflicts with preconceptions.

[DrVanNostrin]

I just noticed my math was wrong. There are only 6 ways to have KK. 3 ways to have AA before the flop and 1 way after the flop.

However, the concept remains the same.

[planoj]

Try explaining it to him like this: "Imagine you hold AA and the flop is AA5. Is there still a 15% chance that your opponent has an Ace?"

[Jouster777]

Love it!