I'm very excited to announce the launch of my latest strategy pack which continues to explore theory of bet sizing and how to use GTO calculations to estimate optimal bet sizes. The pack is available for purchase in the GTO dojo here: http://gtorangebuilder.com/#gto-dojo. The first 8 minutes of the video are available for free below.

This pack picks up where Bet Sizing 1 left off with coverage of IP and OOP 3-bet pots, OOP single raised pots, new key strategy lines and more. I go through additional easily applicable board specific bet sizing strategies, measure how much board specific bet sizing strategies shift as hand ranges change and show how key strategy lines such as turn check-raise all ins in 3-bet pots relate to optimal bet size chioces. With over 100 minutes of video, 50+ GTORB flop solutions, and thousands of SPF solution source files this pack contains everything you need to immediately improve your bet sizing.

And through February 29th we're offering a special discount where anyone who buys the new GTO Bet Sizing 2 strategy pack can get $50 off any one other strategy pack of their choice. Just purchase both packs from our website and email gtorangebuilder@gmail.com and I'll refund $50 to you within 24 hours.

Note that the $50 off only applies to strategy packs that I made, so it applies to any pack in the GTO Dojo with the exception of the the Spins/HUSNG packs.

## Wednesday, February 17, 2016

## Monday, February 15, 2016

### GTO Brainteaser #10 Solution

In our most recent brainteaser, here, I posed a probability puzzle about the relative likelihood of seeing HHT or HTT first in a sequence of coin flips. I won't restate the full problem so please check out this post for the details: http://blog.gtorangebuilder.com/2016/02/gto-brainteaser-10-flipping-coins.html. Its been a while since I did a brainteaser so if you enjoy this type of content and want to see more let me know in the comments below.

So, how do we go about solving this type of problem?

The key concept is that we will state things interms of the EV of reaching a specific state. Since many sequences of flips lead us back to a state that is EV equivalent to our starting state this lets us set up an equation in terms of our starting EV that we can solve. Thinking of the possible transitions from one EV state to another is a core concept in working out complex sequential EVs and is relative in situations like estimating your odds of winning a HUSNG vs an opponent when the stacks are not equal.

Lets begin by computing the EV of our coin flipping game which requires determining the probability with which our sequence ends at HHT or HTT. We should note that to get HTT without getting HHT first requires that prior to or first H we had either just started the game, or we had flipped a T.

Lets call S (S for start) the state where in our most recent history of flips neither of our last two tosses are H and the game has not ended. Call the EV of this state E. Note that the game starts in state S so (we have no previous tosses so they cannot be H) so E is also the EV of the game as a whole.

Now what can happen from state S? If we toss and get tails we stay in state S. If we toss and get H we move into a new state which I will call P1.

So we can already write the equation

E = 1/2 * (EV of P1) + 1/2 * E.

If we subtract 1/2 * E from both sides and multiply both sides by 2 we get

E = EV of P1

Now lets consider what can happen once we reach P1.

1/2 * (EV of P1) + 1/2 * Win

EV of case 2 = 1/2 * EV of case 2 + 1/2 * Lose.

Note that if we substract 1/2 * EV of case 2 from both sides and multiply by 2 this simplifies to

EV of case 2 = Lose

Since case 1 and case 2 are equally likely this means that:

EV of P1 = 1/2 * (1/2 EV of P1 + 1/2 Win) + 1/2( Lose)

This simplifies to

3/4 EV of P1 = 1/4 Win + 1/2 Lose

Multiply both sides of the equation by 4/3 to get EV of P1 = 1/3 Win + 2/3 Lose

Which means our original Equation can be simplified to

E = 1/3 Win + 2/3 Lose.

This means that HHT comes first 2/3rds of the time and the EV of the game is 1/3 * 105 - 2/3 * 100 = -31.6666. This is a very profitable game for the casino despite it seeming at first glance like HHT and HTT should happen equally often!

I'll leave the solution to the bonus as an exercise for those of you are excited to get your hands dirty with some math but I will state the final result which is quite surprising.

It terms out the despite the fact the HHT comes first 2/3rds of the time, on average it takes more flips to get HHT than to get HTT. This seems like a paradox, when we win the game it happens more quickly on average than when we lose but we lose more often than when we win. I suspect that if a casino were to actually provide this game that paradox would lead many to naively overestimate their chances of winning as they played.

I know it can also be helpful to see the results of these types of problems via simulation as well as by mathematics so I've put up some free runable code that monte-carlos the result here: https://jsfiddle.net/sg9g33t2/

### Solution

So, how do we go about solving this type of problem?

The key concept is that we will state things interms of the EV of reaching a specific state. Since many sequences of flips lead us back to a state that is EV equivalent to our starting state this lets us set up an equation in terms of our starting EV that we can solve. Thinking of the possible transitions from one EV state to another is a core concept in working out complex sequential EVs and is relative in situations like estimating your odds of winning a HUSNG vs an opponent when the stacks are not equal.

Lets begin by computing the EV of our coin flipping game which requires determining the probability with which our sequence ends at HHT or HTT. We should note that to get HTT without getting HHT first requires that prior to or first H we had either just started the game, or we had flipped a T.

Lets call S (S for start) the state where in our most recent history of flips neither of our last two tosses are H and the game has not ended. Call the EV of this state E. Note that the game starts in state S so (we have no previous tosses so they cannot be H) so E is also the EV of the game as a whole.

Now what can happen from state S? If we toss and get tails we stay in state S. If we toss and get H we move into a new state which I will call P1.

So we can already write the equation

E = 1/2 * (EV of P1) + 1/2 * E.

If we subtract 1/2 * E from both sides and multiply both sides by 2 we get

E = EV of P1

Now lets consider what can happen once we reach P1.

**Case 1**: Our next flip is T -- If we get a tail then we are at least 3 flips from getting HHT, On our third flip 2 things can happen. Either we get H and are back in state P1 or we get T and the game ends and we win. So the EV of case 1 is just1/2 * (EV of P1) + 1/2 * Win

**Case 2**: Our next flip is a H -- If we then get a 3rd head in a row we are back in Case 2 where our most recent history is HH. If we get a Tail as our 3rd flip the game ends and we lose. Thus the EV of case 2 is justEV of case 2 = 1/2 * EV of case 2 + 1/2 * Lose.

Note that if we substract 1/2 * EV of case 2 from both sides and multiply by 2 this simplifies to

EV of case 2 = Lose

Since case 1 and case 2 are equally likely this means that:

EV of P1 = 1/2 * (1/2 EV of P1 + 1/2 Win) + 1/2( Lose)

This simplifies to

3/4 EV of P1 = 1/4 Win + 1/2 Lose

Multiply both sides of the equation by 4/3 to get EV of P1 = 1/3 Win + 2/3 Lose

Which means our original Equation can be simplified to

E = 1/3 Win + 2/3 Lose.

This means that HHT comes first 2/3rds of the time and the EV of the game is 1/3 * 105 - 2/3 * 100 = -31.6666. This is a very profitable game for the casino despite it seeming at first glance like HHT and HTT should happen equally often!

I'll leave the solution to the bonus as an exercise for those of you are excited to get your hands dirty with some math but I will state the final result which is quite surprising.

It terms out the despite the fact the HHT comes first 2/3rds of the time, on average it takes more flips to get HHT than to get HTT. This seems like a paradox, when we win the game it happens more quickly on average than when we lose but we lose more often than when we win. I suspect that if a casino were to actually provide this game that paradox would lead many to naively overestimate their chances of winning as they played.

I know it can also be helpful to see the results of these types of problems via simulation as well as by mathematics so I've put up some free runable code that monte-carlos the result here: https://jsfiddle.net/sg9g33t2/

## Thursday, February 4, 2016

### GTO Brainteaser #10 -- Flipping coins

While this problem isn't directly related to game theory, it does bring up some very interesting concepts from probability which is another essential mathematical topic in poker. Its also a really great brainteaser so I decided to go ahead and post it even if it is 100% GTO related. Disclaimer, I didn't not make this problem up, its a very old problem often taught in statistics that I was recently reminded of by reddit.

You go to a casino and see that they've introduced a new game.

They are flipping a fair coin and to play you must make a $100 bet. They will start flipping the coin and keeping track of the outcomes of heads or tails. They will keep flipping until they either get the sequence heads, heads, tails or heads, tails, tails over their last 3 flips.

If heads, heads, tails comes first the game ends and you lose your bet of $100 for a loss of $100. If heads, tails, tails comes first the game ends and you win $105 (plus you get your $100 back) for a profit of $105.

So for example if they get:

HTHT

THT

Is the game positive EV? What is the EV of the game?

Also, just incase you missed some of my past brainteasers here are links to the past problems (except the true/false quizzes).

**Problem**:You go to a casino and see that they've introduced a new game.

They are flipping a fair coin and to play you must make a $100 bet. They will start flipping the coin and keeping track of the outcomes of heads or tails. They will keep flipping until they either get the sequence heads, heads, tails or heads, tails, tails over their last 3 flips.

If heads, heads, tails comes first the game ends and you lose your bet of $100 for a loss of $100. If heads, tails, tails comes first the game ends and you win $105 (plus you get your $100 back) for a profit of $105.

So for example if they get:

HTHT

**HTT****Then you win. On the other hand if it comes**

THT

**HHT****then you lose.**

Is the game positive EV? What is the EV of the game?

**Bonus**: On average, when you win how many flips does this game take? What about when you lose, what is the average number of flips in that case?Also, just incase you missed some of my past brainteasers here are links to the past problems (except the true/false quizzes).

- GTO Brain Teaser #1: Exploitation and Counter-Exploitation in Rock Paper Scissors
- GTO Brainteaser #2 -- Red or Black
- GTO Brainteaser #3 -- Blind Man's Bluff
- GTO Brainteaser #5 -- Bayesian Werewolves
- GTO Brainteaser #6 -- Barreling and Bluffing
- GTO Brainteaser #8 -- Solving the Turn, Theory vs Practice
- GTO Brainteaser #9 -- Multistreet Theory: Range Building with Draws

## Monday, February 1, 2016

### New SimplePostflop HUSNG Preflop Solution Pack is Available

I'm excited to announce that I've worked with SimplePostflop to release a preflop strategy pack that has HUSNG solutions at every stack size from 5BB up to 25BB in 1BB increments. All the solutions use 184 board flop subsets, include overbets and standard bet sizes in the postflop game tree, and have very low nash distances, making the pack for more precise than existing approximations.

You can learn more and buy the pack here: simplepostflop.com/en/Solutions_Library?source=gtorangebuilder. By using that link you will also get a $17.50 discount. You need to download the free version of SimplePostflop to view and browse the preflop solution files, but no other license is required, you can just buy the pack and you're good to go.

I believe this is the most comprehensive HUSNG solution pack out there and the results are pretty fascinating. Some highlights of interest:

You can learn more and buy the pack here: simplepostflop.com/en/Solutions_Library?source=gtorangebuilder. By using that link you will also get a $17.50 discount. You need to download the free version of SimplePostflop to view and browse the preflop solution files, but no other license is required, you can just buy the pack and you're good to go.

I believe this is the most comprehensive HUSNG solution pack out there and the results are pretty fascinating. Some highlights of interest:

- Between 18bb and 25bb stacks we see a slow steady shift from limping to minraising from the SB, with the limping frequency decreasing about 3% for each 1bb increase in effective stack size. At the same time the SBs EV advantage steadily grows as stack depth increases.
- At stack sizes of 20bb+ a 2.5x raise against a limp is almost never used and a large size is favored. However in the 12-16bb range a mix of using two different raise sizes against a limp is favored.
- 13bb is about the lowest stack size at which the SB should ever use a minraise as part of his strategy.

And of course there is much more. To fully study and learn the full GTO strategy from this pack would take weeks or months and it contains everything you need to implement GTO play in HUSNG.

I've also gone ahead and put up 5

I've also gone ahead and put up 5

**free**GTORB postflop solutions that use the minraising and calling ranges from the preflop solution pack at 25bbs. You can explore them at the links at the bottom of the post.
I've also found it very interesting to compare the results in this pack to the results from Coffeeyay's SB vs BB Spins pack (where the SB is OOP) as it allows us to measure the value of position and to analysis its effects. That pack is also available here and GTORB users can get $10 off with this link: http://simplepostflop.com/en/Solutions_Library?source=gtorangebuilder

**Free GTORB Postflop Solutions**

**http://gtorangebuilder.com/#share_calc=Ac7h5s_363f5b8268a0b631acaa60fab93c9a89**- http://gtorangebuilder.com/#share_calc=KcJd8d_79b20a354790fd146dd1bc8e9f7816e8
- http://gtorangebuilder.com/#share_calc=AcQh2s_935bbac07318b9782429c4eef5bae75b
- http://gtorangebuilder.com/#share_calc=5h8h2c_77f6662470aa059dc694a5af2082a02f
- http://gtorangebuilder.com/#share_calc=7d7sQc_0813687049c288856569c9f74792847b

Enjoy!

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