I basically understand insurance. Betting against yourself when you're the favorite in case you get sucked out on. But I don't get the math of it. Example - Tony G and Phil Hellmuth. Hellmuth has nines to Tony's AK. They get it all in. Before the river Phil gets insurance from Tony. 12K with 10% juice. Tony hits the A on the river, and ends up owing Phil 70K for the insurance. I don't understand how they come up with that total. Oh, and the pot was 108K.

Your payout is based on your equity at the point of the insurance.

In this hand, Tony had ~14% equity on the turn (correct me if I'm wrong, I watched the episode a few days ago, flush/straight draws would increase the %). That would result in a ~7.14 (100/14) times payout.

Since there's 10% juice, the actual amount bet is for $12,000/1.1 = $10,909

Total Juice = $12,000 - $10,909 = $1,091

Therefore, there can be two results (0% chance of a split pot):
1. Tony loses the hand:
-loses $50k+ put into the pot
-loses $1,091 for juice
-wins $10,909 from insurance

2. Tony wins the hand:
-wins $50k+ from the pot (108k total, only half is profit)
-loses $1,091 for juice
-loses $10,909 x 7.14 = ~$77,921

Phil's $70k was an approximation, as he did the math in his head. As you can see, the 10% juice that Hellmuth collects guarantees this is a profitable proposition.

Lol........ Very nice +1000!

Wow. I guess there is a reason I didn't take any math higher than Alg 2. Thanks for the explanation, but it still goes over my head. But that's ok. However, what I don't get is why would Tony give Phil insurance in that scenario? By winning the pot, he actually loses money if he owes Phil over 70K. His profit from the hand was only 50K or so.

I'm not exactly sure who gets the juice in this hand. It actually make sense for the juice to go to Tony, and is like Phil's "payment" to Tony to give him an added incentive to take a -EV deal.

Theoretically, if there was 0% juice, no matter what amount you insure (as long as the larger payout doesn't exceed the size of the pot), both parties reduce variance. In this hand, the $12k that Phil offered happened to exceed the size of the pot in Tony sucked out.

I guess the new values should be:

1. Tony loses the hand:
-loses $50k+ put into the pot
-wins $1,091 for juice
-wins $10,909 from insurance

2. Tony wins the hand:
-wins $50k+ from the pot (108k total, only half is profit)
-wins $1,091 for juice
-loses $10,909 x 7.14 = ~$77,921

If you want to calculate EV for this hand, we can compare with insurance and without insurance.

Without Insurance:
-Tony wins 14% of the time, Phil wins 86% of the time. Pot = $100k for simplicity purposes.

Tony's EV = (14% x $50,000) + (86% x -$50,000) = $-36,000
Phil's EV = (14% x -$50,000) + (86% x $50,000) = $36,000

With Insurance:

-Tony wins 14% of the time, Phil wins 86% of the time. Pot = $100k for simplicity purposes.
-Tony wins $1,091 100% of the time.
-Phil loses $1,091 100% of the time.

Tony's New EV = Juice Earned + EV in Hand + EV from Insurance
Juice Earned = 100% x $1,091 = $1,091
EV in Hand = (14% x $50,000) + (86% x -$50,000) = $-36,000
EV from Insurance = (14% x $-77,921)+ (86% x $10,909) = -1,527

Total = $-36,436

Phil's New EV = Juice Paid + EV in Hand + EV from Insurance
Juice Paid = 100% x $-1,091 = $-1,091
EV in Hand = (14% x -$50,000) + (86% x $50,000) = $36,000
EV from Insurance = (14% x $77,921)+ (86% x $-10,909) = $1,527.2

Total = $36,436.20

to the nth degree. Dude, your calculations are impressive, but I don't have the foggiest in how to follow them. Thanks anyway.

I think I disagree with how you have calculated insurance. If you listen to what they said, Phil said, "I'll GIVE you 10% juice. If I win, you get $12,000, if you win, I get a lot more."

Tony G had 6 outs of the remaining 44 "unknown" cards, so odds against were 38 to 6.

Insurance with no juice:
If the favorite (Phil) wins the pot, he pays the dog $12,000 (that's 2x6)
If the dog (Tony) wins the pot, he pays the favorite $76,000 (that's 2x38)

Let's check this to see if it is right - run it 44 times and have it come out just like the odds would predict:
Favorite wins 38 times x $12,000 totals $456,000
Underdog wins 6 times x $76,000 totals $456,000
Perfect.

Phil agreed to "Give" Tony 10% juice but that really meant that if he (Phil) lost the pot and won the insurance, Tony only needed him to pay the full amount less 10%.

$76,000 less 10% is $68,400

So in the end, Tony should have won a pot of $108,000 but had to pay Phil $68,400 which was more than what Phil put into the pot. A great deal for Phil. The insurance protected Phil's pot equity.

Pot equity:
Pot was $108,000
Phil was 38/44 to win (13.63%), so $108,000 x 38/44 = $93,272 of "Pot Equity".

Insurance reduces variance. How:
Without insurance, if Phil wins he gets $108,000. Lose and get zero.
With insurance, if Phil wins he gets $108,000 less $12,000 for insurance = $96,000
With insurance, if Phil Loses he collects the $68,400 from Tony.

BTW Phil started that hand with $53,500, so by getting insurance, he assured himself of a profit regardless of how that river card came out.

By "Selling" insurance to Phil, Tony essentially gave up on the idea of winning the pot and was willing to accept $12,000 if he lost it as a big underdog and was willing give most of it back to Phil if he got lucky and won it, so Tony guaranteed himself a loss on the hand but a smaller one than the likely outcome.

Note: All of this is my idea of what I think is happening when Hellmuth buys insurance and how I think they work out the math. If my line of reasoning is incorrect and they don't really do it this way, please let me know.

Note 2: Phil told Tony at the end that he owed him about $70,000.