Converting probability to odds, problem with a tied result

[purp]

Hi

I'm having difficulty converting a hands probability of winning into odds because I don't know how to include the probability of a tie in those odds.

This is in theory rather than having a practical use, let me explain the senario:

I have 3 hold'em players each with the pocket cards face up, I want to offer fair odds to a 4th person who can bet on which hand will win. The 4th person can only bet on a hand winning and not the tie. If a tie happens I would divide the odds by the number of tied players. So if he bet on hand 1 at odds of 4.00 and all 3 hands tied, he would get odds of 1.33.

While converting the probabilities to the odds I offer is not a problem (when excluding the probability of a tie), I'm not offering a tie bet so I have to factor these into the odds somehow.


Example:
Player 1 holds 8d, 7d
probability of win = 29.49%
Probability of tie = 2.17%

Player 2 holds 2s, Ah
Probability of win = 42.16%
Probability of tie = 0.34%

Player 3 holds 8c, Kh
Probability of win = 26.18%
Probability of tie = 2.17%

I convert the win probability to decimal odds by dividing 100 by the probability so:

Player 1 = 100/29.49 = 3.39
Player 2 = 100/42.16 = 2.37
Player 3 = 100/26.18 = 3.82

The above decimal odds would be a fair reflection of the the hands chances of winning at this stage, however they don't factor in the tie.

Anyone have a simple answer to help me solve this?
Thanks in advance
Purp

[filsteal]

Just to clarify:

He bets on Hand X at odds of, say, 4-1.

If Hand X ties for the pot with Hand Y, he gets 2-1, and if hand X ties for the pot three-way with Hands Y and Z, he gets 1.33-1. Is that correct?

[filsteal]

Also, your method for converting probabilities to odds is wrong.

If a hand wins 50% of the time, its odds are 1-1, not 2-1.

You need to subtract 1 from your answers. For example:
Player 1 = 100/29.49 - 1 = 2.39

[purp]

Hi filsteal,

In answer to your 2nd reply, I was using decimal odds in the example, so 2.00 is 1:1 and 4.00 is actually 3:1. Decimal odds include the stake in the return.

On the 1st reply, yes the odds would be split as you said apart from the confusion around decimal and fractional odds. It's the decimal odds that would be divided by the number of tied players.
So if the odds were 4.00 (or 3:1 expressed in fractional odds) and it was a 2 way pot his decimal odds would be divided by 0.5 so he would get 2.00 (or evens), if it was a 3 way pot his decimal odds would be divided by 3 so he would get 1.33 (or 1/3 fractional odds).

I actually think using equity calculations would solve my problem anyway. Equity calculations will show me the % of the pot that a hand wins on average over many trials, including the tied pots.
As the equity of a hand is calculated by dividing the number of "pots" that the hand won by the number outcomes considered.
If a hand can't win it would have zero probability of winning, but it will have some small equity because it may be able to tie.

[filsteal]

My bad on decimal vs. fractional odds. I wasn't familiar with the term

And yeah, you can just take 100/%equity, and then everything will work fine.

[purp]

Ok thanks, equity it is then [img]/images/graemlins/grin.gif[/img]


BTW I made an error here
[ QUOTE ]

So if the odds were 4.00 (or 3:1 expressed in fractional odds) and it was a 2 way pot his decimal odds would be divided by 0.5 so he would get 2.00 (or evens)

[/ QUOTE ]
Should say divided by 2 not 0.5