Poker problem

[Raydain]

This may be an easy problem but it was bugging me a lot last night.

Hypothetical: Let's say you're playing against an opponent that always follows strict guidelines. Villain raises preflop every single hand but will only make continuation under the right cirucumstances.

Villain will make a continuation bet if he hits the flop or if his last continuation bet was not a bluff.

First hand: Hero sits down at Villian's left. Hero observes Villian raise preflop and make a continuation bet on the flop to take it down.

Second hand: Villain raises preflop and makes a continuation bet.

Assuming Villain hits the flop 1/3 times, what are the odds that villain is bluffing?

[Siegmund]

I am sure there's more than one way to interpret your question, but I translate the following:

[ QUOTE ]

Villain will make a continuation bet if he hits the flop or if his last continuation bet was not a bluff.


[/ QUOTE ]

to mean "Villain will make a continuation bet if he hit the flop on either of the last two deals." That is, he bets every time he hits the flop, and again the first time he misses as a bluff, but never bluffs twice in a row.

If that's the case, Villain will hit this flop with probability p, but bet with probability 2p-p^2. So, the odds of hit the flop vs. bluff are 1 : (1-p).

Taking p=1/3, that means 2/5ths of this villain's continuation bets are bluffs when he has also bet the previous hand (and none of them when he hasn't - since the only way to make him check is for him to have missed the flop last time AND bluffed and failed before that.)

[Chamonyx]

40% isn't it?

Over 2 hands, possibilities are HH,HM,MH,MM with probabilities 1,2,2,4 over 9 (Hit, Miss)

Only case he doesn't bet on second hand is MM
Only case he bluffs on second hand is HM

P(Bluff/Bet) = P(Bl & Bet)/P(Bet) = 2/5

[sthief09]

if we can assume he bet the last hand regardless of whether he hit the flop, since he's new, then it was a bluff 2/3 of the time. of that 2/3 which was a bluff, now 1/3 of the time he'll bet and 2/3 he'll check. so 2/9 of the time he bets, which is a hand. now if he hit the flop last hand, which is 1/3 of the time, he'll bet 1/3 fo this time with a hand, for a total of 1/9, and he'll bet as a bluff 2/3 of that time which is 1/3*2/3 which is 2/9. so 3/9 of the time he bets a legitimate hand, 2/9 of the time as a bluff, and 4/9 of the time he'll check. so it's a bluff 40% of the time

[Raydain]

[ QUOTE ]
if we can assume he bet the last hand regardless of whether he hit the flop, since he's new

[/ QUOTE ]

He's not new. We just sat down to his left at an already active table.

[bobman0330]

I'm coming up with a slightly different answer:
It's important to consider the last THREE hands, because that determines villain's action if he missed last hand and hit this hand.

All 8 possibles:
ways(of 27); last 3 hands; last 3 betting actions
1; HHH; BBB
2; HHM; BBB
2; HMH; BBB
4; HMM; BB-
2; MHH; ?BB
4; MHM; ?BB
4; MMH; ?-B
8; MMM; ?--

So, 11/27 of the time (HHH; MHH; HMH; HHM; MHM), he will have bet the last 2 hands. 16/27, he does not (MMM; HMM; MMH). Of the 11/27, he is bluffing only 6/27 of the time (HHM; MHM). that's 6:5 in favor of bluffing, or 6/11.

[Raydain]

can we get a consensus on the answer?