How Gus Hansen came up with 30%?

Question

Taken from Every Hand Revealed.

Here's the scenario:

Gus hand was A♣ 4♣

It was mentioned that the flop was 4-handed.

The flop was:

2♥ 5♦ K♣

A bet was made, Gus and one villain stays.

The road was:

J♣

Villain bets $2,200. Pot is $5,200. Villain remains with 7k.

Here's the question.

I have to call $2,200 to win $5,200. Which means I need aprox. 30% winning chance

-Gus Hansen

The question is, why 30%?

Answer 1

Your question is a great example of some of the math that you need to carry out at the table.

The first part is to know the value you're getting from the situation. Since Gus is calling a smaller amount than what's in the pot, there's an overlay available. To do this you take the pot divided by the amount you have to call: 5200/2200 = 2.36. You can express this by thinking "I am getting 2.36-1 odds on my money". Sometimes you hear players say, "The pot was laying me 2-1", or something along those lines.

Next, you need to translate this into a percentage that gives you a break-even value. If you won an overlay of 2.36 every time, you would be a millionaire in no time, if you won it once every 10 times, you would be broke in no time, depending on the probability that you're right of course! Defining your opponents true range as close as possible will provide that answer.

Perhaps the quickest way to get to a number is to add a 1 to your overlay number and change it into "decimal odds": 2.36 + 1 = 3.36

To understand how often you need to break even, you would divide 1 by the decimal odds. In this case you need to win about 29.8% of the time (1 / 3.36 = 0.298) of the time. That's close to 1 in 3 tries. Hanson calculated he'd need to win around 30% of the time or more to make calling profitable if he got to showdown with no extra betting.

Lets use $1 and your overlay is $2.36 - so now iterate it out as if you were gambling:

First try: lose $1

Second try: lose $1

Third try: win $2.36

At the end of 3 tries, you would be netting a win of $0.36

Or, to use Hanson's example:

First try: lose $2200

Second try: lose $2200

Third try: win $5200

At the end of 3 tries, you would be netting a win of $800

This is one of the ways that you can evaluate if a move is +EV. Of course, now you have to decide if your hand will actually win 30% of the time. If you're wrong, then its possible you made the right decision, but didn't get the outcome you hoped for this time. Another possibility, which could mean folding more often or perhaps raising is better, is that perhaps you misjudged the strength or weakness of your opponents range. This starts to crossover into questions about showdown value and is beyond the scope of your question.

The question "How do I Calculate Expected Value of Shoving, including Fold Equity, in heads up play?" covers the maths of this problem in fine detail.

Hope that helps!

Answer 2

I would suggest you have a read up on Pot Odds

Here are a few sites:

Pot Odds Wikipedia

The Poker bank pot odds

Hopefully these will help you understand.

Answer 3

Thanks to over-sophisticated answers like the ones given here it took me forever to understand this stuff.. It's actually way easier to calculate: bet/pot+bet -> 2,200/5,200+2,200=0.297 which translates to 30% "pure" pot odds. So you'd need at least a 30% winning chance to profitably call here.