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Buzz,Do you agree with these numbers?
[/ QUOTE ]Hi Ben and Corn Husker Fan - I've been in Las Vegas for the past ten days, away from my computer, and not able to answer your question until now.
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So the probability of catching good on both the turn and the river is 0.444 x 0.364 or 0.162.
[/ QUOTE ]Yes. That is correct.
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I used Alspach’s Poker math to estimate this probability at 0.3 (<
http://www.math.sfu.ca/~alspach/mag8/>)
[/ QUOTE ]Alspach is the mathematician, not me. However, that estimation (for getting quartered and sixthed) is incorrect (unless Corn Husker Fan is referring to a game where only seven players are dealt cards).
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Therefore there is a 0.7 chance of winning ½ the pot and a 0.3 chance of wining a ¼ of the pot. So, if I catch runner runner low, I expect to get ((0.7 x ½) + (0.3 x ¼)) or 0.425 of the pot.
[/ QUOTE ]That's incorrect for a full game.
If (1) Hero is dealt a hand with exactly one ace and exactly one deuce, and with no back-up for low, and (2) when the flop has only one low card (not an ace or a deuce), and (3) if nobody folds the runner-runner nut low draw, and if the turn and river are both cards such that Hero's runner-runner low draw is successful, then<ul type="square">when ten players are dealt cards,
~57% Hero does not split low,
~37% Hero gets quartered for low,
~5% Hero gets sixthed for low, and
~1/1000 Hero gets eighthed for low.
when nine players are dealt cards,
~61% Hero does not split low,
~34% Hero gets quartered for low,
~4% Hero gets sixthed for low, and
~1/1000 Hero gets eighthed for low.
when eight players are dealt cards,
~65% Hero does not split low,
~31% Hero gets quartered for low,
~3% Hero gets sixthed for low, and
~1/1000 Hero gets eighthed for low.[/list]Something like that.
However, it's a moot point because strong players will probably not be drawing exclusively to runner-runner low. Thus if some fool draws for runner-runner low, he'll probably not have strong players sticking around to possibly quarter him. And that makes it impossible to compute how often Hero will get quartered or sixthed when drawing for runner-runner low.
Hypothetically we can create a situation where enough players call three or four bets on the first betting round and where there subsequently is no more raising on the second and third betting rounds, so that Hero has favorable odds to draw to a runner-runner low. However, my experience is after lots of action on the first betting round there is generally raising on the third betting round, if not on the second.
It's not absolutely impossible that Hero could have odds to continue to a bet - but after an action packed first betting round, even if there's no raising on the second betting round, how could Hero ever be sure there would not be raising before the river? And Hero would not have favorable odds to continue if there was more raising before the river.
Figure that if the probability of Hero making the runner-runner low draw is 0.162, then roughly Hero makes the draw one time out of six and misses the other five times out of six. When Hero misses, Hero misses roughly half the time on the turn (cost one more small bet, assuming there are no raises) and half the time on the river (cost one more small bet plus one large bet, assuming there are no raises). (It's worse for Hero with raises). Roughly (assuming there are no raises on the second or third betting rounds) Hero's cost averages two small bets = one big bet when he misses. And he misses five times out of six.
Hero has to make up the cost of those five misses when he doesn't miss - but that's unlikely. Hero's share of the pot will simply not average enough to make up for his costs when he misses. In other words, paying to draw exclusively for runner-runner low is simply not a good bet.
Assuming no raises, even without getting quartered or sixthed, Hero has to win at least five big bets (ten small bets) when he makes his runner-runner draw. That's realistic for scooping the pot, but not for splitting the pot. It's not impossible - just not realistic.
If I get a free draw, whoopee! One time out of six it might work out well for me - and even better than just getting half the pot. Sometimes you can turn a low winner into a scooper by just betting the low.
However, even taking into account that I might be able to convert a low hand into a scooper by aggressive betting, totally missing high on the flop and needing a runner-runner for low is a fairly standard check/fold for me. (I generally check, hoping for a free card, but fold when there is a bet).
I'm not saying that drawing to runner-runner low is something one should absolutely never do, but I think drawing to runner-runner low with no realistic chance for high is a fairly common mistake.
Buzz