This is a discussion on Low Board Probability within the online poker forums, in the General Poker section; Any of you who are actually interested in the math behind poker may be familiar with the writings of Brian Alspach, Professor Emeratus In Graph 


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Low Board Probability
Any of you who are actually interested in the math behind poker may be familiar with the writings of Brian Alspach, Professor Emeratus In Graph Thoery from Simon Fraser University in B.C. http://www.math.sfu.ca/~alspach/index.html. It's nice to know that there are true geniuses in the world who share our unhealthy addiction to this feckless yet engrossing game.
I emailed Professor Alspach for permission to use some of his schedules and tables for the benefit of our forum and he responded almost immediately with approval, provided that credit be given where due. Well, as much as I would like to take credit for it, I have no problem with this restriction as nobody in their right mind could possibly mistake his brilliant work for my own anyway. So, Without furthar adieu, here is the first table that caught my eye as both interesting and particularly practical for immediate use. Many of us like to mix it up preflop with a pocket pair. And why not when 1 in 8 times your Mickey Mouse will turn into Mighty Mouse. But what about those other seven times when Christmas doesn't come early? What's your plan? Well, if your stepping into a pot with the hopes of bulldozing the felt with top pair or overpair, you really should have some idea of how likely that is. It should come as no surprise that a whopping 100% of the time, your ducks will have overcards and your bullets won't. We don't need the good professor to tell us that do we? How 'bout The cowboys? If you look at the chart below, you will see that nearly 25% of the time the flop will present at least one overcard, an Ace of course. WHAT! How can this be? There's only one rank higher than my kings and I'm only safe 75% of the time? No way, this site is juicing the deck. But it get's worse. CantBeBeat is sitting on the button when UTG TotalRock raises 3xBB . He looks down and is thrilled to see a pair of ladies. Not bad. CantBeBeat could reraise or go allin, but he doesn't want to scare TotalRock away so he opts to just call and wait for the Rock to bet out on the flop. The SB folds and LovesToBluff, the BB, calls the 2 bets. The flop comes up 8KJ. Bluff bets out and Rock folds. Now what? As TeddyKGB says in Rounders, "Didn't expect that did you?". But you should have. The table below shows that over 40% of the time the flop will present at least one overcard. Scary isn't it? Here's how it works. And you will all please excuse me if I allow Professor Alspach to explain it in his own words. "There are 50 cards he cannot see. We are choosing three cards from 50 for the flop, so there are C(50,3) = 50!/3!47! = 19,600 flops. How many of the flops contain at least one ace or king? The easiest way to determine this is to count the number of flops containing none of them and subtract it from 19,600. Of the 50 unseen cards, 42 remain if we disallow aces and kings. Thus, there are C(42,3)=11,480 flops containing neither an ace nor a king. Hence, there are 8,120 flops containing at least one ace or king. Then the probability of an overcard coming in the flop is 8,120/19,600 = 0.414. This undoubtedly would surprise the losing player and he would interpret it as an unintuitive result." Mind bending stuff huh? But really, anyone with a $10 scientific calculator and the desire to work it out for themselves can reproduce this. Well, I've already spent more time on this than I had intended and the kids are screaming for story time so I'll make my exit here. I suggest anyone who'e interested in more of this visit the site linked above. I hope you don't mind Nick. pair board 2,2 1.00 3,3 0.999997 4,4 0.99988 5,5 0.9991 6,6 0.996 7,7 0.988 8,8 0.969 9,9 0.933 10,10 0.869 J,J 0.763 Q,Q 0.599 K,K 0.353 A,A 0.00 Note: The top number represents the probability of an overcard on the flop and the bottom number represents the probability of an overcard on the board 