Theory - Scare cards and inducing bluffs math check up

kitaristi0 · #1 09-01-2006, 03:20 AM

Just something I scribbled down while on the train today. Can someone check my math? I know that the EV(check) equation isn't complete because it's missing the times he hits a set or a K and the times that an overcard to his pair hits, but whatever. Am I right so far?

UTG limps $2, CO raise to $9, you reraise to $32 with black aces. Co call, Pot $69 (We both have $168 behind)

The flop is T92 2-flush (two spades)

When villain calls your reraise his range is JJ+/AK. There are 6 combos of JJ,QQ and KK. 1 combo of AA, and 8 combos of AK. Thus it's 19:8 that he has an overpair. Assume that on the flop whenever he has an overpair he is going to CRAI. You will call and win 90.9% of the time. If he has AK and you bet he will fold. The EV of betting is:

EV(bet) = (19/27)(.909)($237) + (19/27)(.095)(-$168) + (8/27)($69)
EV(bet) = $151.60 + (-$11.23) + $20.44
EV(bet) = $160.81

Assumptions if you check the flop:

- Villain will bet $50 with all of his holdings. Then you raise all-in. If he has an overpair he will call. If he has AK he will fold.
- Assume for now that the turn is always a blank (see below)
(NOTE TO SELF: Figure out the probability of an overcard to his pair hitting on the turn and factor that into the calculations)
- However, if the turn is a spade villain won't call all-in with his overpairs unless he has a spade in his hand. The odds of a spade hitting is (10/45). The odds of a spade not hitting are (7/9).
- The odds of him having a spade in his hand is (1/2)

EV(check) = (7/9)(19/27)(.954)($237) + (1/2)(10/45)(19/27)(0.954)($237) + (1/2)(10/45)(19/27)($50) + (8/27)($50) + (7/9)(19/27)(.095)(-$168) + (1/2)(10/45)(19/27)(0.095)(-$168)

EV(check) = $123.75 + $17.68 + $3.91 + $14.81 + (-$8.73) + (-$1.25)
EV(check) = $150.17

Note that this number is actually lower because some of the time an overcard to villain's pair will hit and he won't donate his whole stack. Also some of the time he will make a set on the turn. This however is somewhat offset by the times he turns a K and gets stacked with AK. Also some of the time he won't pay off his whole stack even though he has a spade in his hand, e.g. if he has JsJd he may not call a shove on with just the 4th nut draw.

Leviathan101 · #2 09-01-2006, 03:27 AM

just a quick note, but is there any reason he wouldn't get stack on the turn with AK if a the 4th ace comes? Why only the King?

Very interesting btw.

kitaristi0 · #3 09-01-2006, 03:29 AM

[ QUOTE ]
just a quick note, but is there any reason he wouldn't get stack on the turn with AK if a the 4th ace comes? Why only the King?

Very interesting btw.

[/ QUOTE ]

Aah yes, obviously an ace would most probably get him stacked too.

Btw, do you play on Full Tilt as leviathin999?

kitaristi0 · #4 09-01-2006, 02:28 PM

Bump for the whatever time it is now crowd. If someone who knows math could look over my equations and tell me whether I'm doing them right that would be great.

Isura · #5 09-01-2006, 03:48 PM

[ QUOTE ]

EV(check) = (7/9)(19/27)(.954)($237) + (1/2)(10/45)(19/27)(0.954)($237) + (1/2)(10/45)(19/27)($50) + (8/27)($50) + (7/9)(19/27)(.095)(-$168) + (1/2)(10/45)(19/27)(0.095)(-$168)

[/ QUOTE ]

Is his equity on the turn still 0.095? I thought it would be less. Otherwise the calculation looks fine.

You should try do and factor in when they turn a set, because that costs a lot to us. Also, there are some shortcuts you could have used to compare the 2 EVs [img]/images/graemlins/grin.gif[/img]

kitaristi0 · #6 09-01-2006, 04:11 PM

[ QUOTE ]
[ QUOTE ]

EV(check) = (7/9)(19/27)(.954)($237) + (1/2)(10/45)(19/27)(0.954)($237) + (1/2)(10/45)(19/27)($50) + (8/27)($50) + (7/9)(19/27)(.095)(-$168) + (1/2)(10/45)(19/27)(0.095)(-$168)

[/ QUOTE ]

Is his equity on the turn still 0.095? I thought it would be less. Otherwise the calculation looks fine.

You should try do and factor in when they turn a set, because that costs a lot to us. Also, there are some shortcuts you could have used to compare the 2 EVs [img]/images/graemlins/grin.gif[/img]

[/ QUOTE ]

Yeah you're right, his equity on the turn would be 0.045 not 0.095.

So it's actually:

EV(check) = (7/9)(19/27)(.954)($237) + (1/2)(10/45)(19/27)(0.954)($237) + (1/2)(10/45)(19/27)($50) + (8/27)($50) + (7/9)(19/27)(.045)(-$168) + (1/2)(10/45)(19/27)(0.045)(-$168)

I think.

And btw, you're such a tease for not telling me the shortcuts.

Pokey · #7 09-01-2006, 05:48 PM

WARNING: MATH AHEAD

You asked for this, so don't blame me for what follows.

EV calculations get tricky when you're speculating ahead to the turn AND putting villain on a range instead of a hand.

For completeness, we're going to COMPARE the EVs of all-in-on-the-flop to the wait-and-see-on-the-turn. Also, we're going to lump hands by category: (a) opponent has AA, (b) opponent has AK, and (c) opponent has all other pairs.

(a) When villain has AA

If we push the flop, our equity is 52.27%. Thus, our expected payout is:

Overall payout when we push the flop:

EV = -$168 + 0.5227*($69 + $168 + $168) = +$43.69.

If we wait for the turn, our opponent will bet $50 and call, but will bet-and-fold if the turn is a spade and we push. (O rly?) That means that 35/45 times we push (winning $34.50 from splitting the $69 dead money) and 10/45 times we win $119.

Overall payout when we wait until the turn:

EV = (1/45)*(35*$34.50 + 10*$119) = +$53.28.

Pushing the flop wins us $9.59 less than waiting until the turn if our opponent holds AA.

(b) When villain has AK

By assumption, if we push the flop, villain folds, and we win $69 every time.

Overall payout when we push the flop: +$69

On the other hand, if we wait for the turn villain will bet $50, and will fold unless he spiked an A or K. That is, 42 times out of 45 he bets and folds, but 3 times out of 45 he calls, drawing with 3.03% equity. Our expected payout:

Overall payout when we wait until the turn:

EV = (1/45)*(42*$119 + 3*(0.9697*($69 + $168) + 0.0303*(-$168))) = +$126.05

Pushing the flop wins us $57.05 less than waiting until the turn if our opponent holds AK.

(c) When villain has JJ-KK

OK, time for the hard one.

Our equity against this range is 89%. We're putting in $168, our opponent is putting in an additional $168, and the pot already has $69.

Overall payout when we push the flop:

EV = -$168 + 0.89*($69 + $168 + $168) = +$192.45

ASIDE: Alternatively, we can say that we win 89% of the time and lose 11% of the time, giving us:

EV = 0.89*($69 + $168) + 0.11*(-$168) = +$192.45

Same result, different calculation method. 

Given that we'll never find a fold in the hand, we can never do better than this; however, we CAN do worse if our opponents are able to escape on the turn. Assume we wait to the turn to make our move.

Now, out of the 45 undefined cards in the deck:

1. If opponent has JJ with no [img]/images/graemlins/spade.gif[/img], there are 10 overcards that cause him to fold and there are 7 other spades (not the J[img]/images/graemlins/spade.gif[/img]) that cause him to fold. 17 fold cards

2. If opponent has J[img]/images/graemlins/spade.gif[/img] J, there are 8 overcards that cause him to fold. 8 fold cards

3. If opponent has QQ with no [img]/images/graemlins/spade.gif[/img], there are 6 overcards that cause him to fold and 8 other non-Q spades that cause him to fold. 14 fold cards

4. If opponent has Q[img]/images/graemlins/spade.gif[/img] Q, there are 5 overcards that cause him to fold. 5 fold cards

5. If opponent has KK with no spade, there are 2 overcards that cause him to fold and 9 non-K spades that cause him to fold. 11 fold cards

6. If opponent has K[img]/images/graemlins/spade.gif[/img] K, there are two overcards that cause him to fold. 2 fold cards

Average number of folding cards, given the equal likelihood of behind dealt any of these combinations:

(1/6)*(17 + 8 + 14 + 5 + 11 + 2) = 9.5 cards

So there is a (9.5)/45 chance that we win $69 + $50 = $119.

There is a 2/45 chance that our opponent spikes his pair; when that happens, we win, on average, 9.1% of the time (spiking an ace or hitting our flush) and lose the other 90.9% of the time. We'll get all-in, so our expected payout here is

EV = 0.091*($69 + $168) + 0.909*(-$168) = -$131.14

The remainder of the time, our opponent is drawing to just over two outs (runner-runner straights about 0.2% of the time, ugh), giving us the win about 95.3% of the time, for an EV of:

EV = 0.953*($69 + $168) + 0.047*(-$168) = +$217.96

Overall payout when we wait until the turn:

EV = (1/45)*(9.5*$119 + 2*(-$131.14) + 33.5*217.96) = $181.55

Pushing the flop wins us $10.90 more than waiting until the turn if our opponent holds JJ-KK.

Conclusion.

Pushing the flop wins us $9.59 less than waiting until the turn if our opponent holds AA. There is 1 combination of cards where this applies.

Pushing the flop wins us $57.05 less than waiting until the turn if our opponent holds AK. There are 8 combinations of cards where this applies.

Pushing the flop wins us $10.90 more than waiting until the turn if our opponent holds JJ-KK. There are 18 combinations of cards where this applies.

Grand total EV from pushing the flop:

(1/27)*(1*(-$9.59) + 8*(-$57.05) + 18*($10.90)) = -$9.99.

Given all of the original assumptions, plus the few I added above for clarity, pushing the flop wins us $9.99 LESS than waiting until the turn.

Feel free to check my numbers (if you can stay awake long enough), but feel VERY free to start arguing about the assumptions.

Dan Bitel · #8 09-01-2006, 06:03 PM

Theres exactly ZERO chance of me reading thru all the maths here, but just thought I'd add a quick:

[img]/images/graemlins/heart.gif[/img] pokey

mosuavea · #9 09-01-2006, 06:04 PM

Pokey,

You are one sick mutha [censored]

kitaristi0 · #10 09-01-2006, 06:10 PM

POKEY I LOVE YOU AND WANT TO HAVE YOUR BABIES!!!!! [img]/images/graemlins/heart.gif[/img] [img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img][img]/images/graemlins/heart.gif[/img]