Pre-Flop favorite win probability

[thylacine]

Amongst all Heads-Up Holdem hands, dealt to the river for a showdown, what is the probability that the Pre-Flop favorite wins?

[mannika]

uuuuh... ???

The preflop favourite is the favourite at the river by whatever it is the favourite by.

If it is a 70-30 favourite, then 70% of the time it will be the best at the river.

[thylacine]

Yes but you need to average the favorite's win probability over all possible hands for the two players.

[jimdmcevoy]

Well, to get the exact answer would take a very long time.

My suggestion for an approximation is to first off disregard suits, it won't make a huge difference (I don't think...).

So now there are 91 hands, 78 hands unpaired and 13 hands paired. But don't forget that getting AK (suited or unsuited) is 4/3 more likely than getting AA, this is true for all unpaired and paired cards. This will have to be taken into account.

Now, for further simplifications, let's say two overcards beat two undercards 64% of the time. I used carplayer.com 's poker odds calculater and it seems that this number is usually between 59% and 69%.

Also let's say that when some one has the highest and third highest card they will win 60% of the time (like A3 vs J9)

And let's say that some one with the highest and lowest card will win 56% of the time (like Q2 vs J9)

And if each player shares a card, the higher kicker will win say 70% of the time (this varies a fair amount)

And a pair vs overcards, a pair wins 54%.

A pair vs one over and under card wins 71%.

A pair vs two under cards wins 85%.

A pair vs under pair wins 82%.

A pair vs an equal card plus and over card (eg. 88 vs K8) wins 70%

And finally a pair vs and equal card plus and under card wins 90%

These are my estimates, they may be a little off but I reckon they are close.

So now you have to count how many times each situation will occour. I am not sure whether you want to include situations where the hands are the same (like AK vs AK), I will ignore them.

So let's do all the situations with a pair in it first.

Number of ways you can get pair vs under pair: 156

Number of ways you can get pair vs two undercards: 286

Number of ways you can get pair vs one over and one under: 143

Number of ways you can get pair vs over cards: 286

But don't forget I will need to 'weight' these later, since it is less likely to get a pair than unpaired cards (as in you are more likely to get AK than AA, 4/3 more likely)

Now for the rest:

Number of ways you get two over cards vs two undercards: 715

Number of ways you get highest and third highest vs second and lowest: 715

Number of ways you get highest and lowest vs second and third: 715

And finally number of ways you get one shared card: 429

But for this last shared card scenario you have to 'weight' it by 3/4 since if one person has a King say it makes it less likely for the other person to have a King.

Woops forgot two more,

A pair vs and equal card plus an over: 78

A pair vs and equal card plus under: 78

I have to 'weight' these last two for two reasons, (3/4) for the pair and (1/2) since if some one has a pair of 8's say, it is 50% less likely than normal the other person will have and 8.

Now for the final answer:

[715*64%+715*60%+725*56%+429*(3/4)*70%+156*(3/4)*(3/4)*82%+286*(3/4)*85%+143*(3/4)*71%+286*(3/4)*54%+78*(3/4)*(1/2)*70%+78*(3/4)*(1/2)*90%]/[715+715+725+429*(3/4)+156*(3/4)*(3/4)+286*(3/4)+143*(3/4)+286*(3/4)+78*(3/4)*(1/2)+78*(3/4)*(1/2)+78*(3/4)*(1/2)]

which is 63.7%

So according to my approximations, if two players go all in blind and they don't have the same hand, be prepared to see a 'bad beat' 36.3% of the time.

[thylacine]

Wow, thanks, that looks like an answer. I thought it would be something along the lines that the favorite is not a massive favorite.

Saaaaay, I don't suppose you'd mind doing the same for OMAHA! [img]/images/graemlins/wink.gif[/img]

[jimdmcevoy]

LOL, for Omaha I can do a quick estimate: between 50% and 90% [img]/images/graemlins/wink.gif[/img]

[thylacine]

Yeah, that's what I though too. Seriously though, I wonder if it could be even less than 55%

[jimdmcevoy]

I don't play omaha often, but from my limited experience I'd guess say 58%. Although something I find interesting is that sometimes you are drawing dead preflop in omaha, like if you hold 2222 and some one has 3369. I see a proposition bet comin on with my friends on this one come to think about it...