how often will the board run out as a wheel
Join Date: Dec 2008
Posts: 1,650
A 2 3 4 5 after the river those are the 5 cards in any order. How often should you see some thing like that. The best i have been able to find is around 1 in 4000 all-ins pre is that close?
Join Date: Nov 2012
Posts: 4,143
Not sure about "all-ins pre", as that could presume we know something about some hole cards. How likely, or with what frequency, do 5 randomly drawn cards make a wheel?
Total possible 5 card hands is 52C5 or 52*51*50*49*48/(5*4*3*2*1) = 2,598,960
Total number of wheels is 4**5 or 4*4*4*4*4 = 1024
Odds of 5 random cards making a wheel = 1024/2598960 = .000394004 or about 1 in 2538
We can check that by looking at the odds of each card as it's drawn:
20/52 * 16/51 * 12/50 * 8/49 * 4/48 = .000394004
The above assumed we are including the 4 steel wheels. If we don't want to count them in our total, then it's 1020/2598960 = .000392 or 1 in 2548
Join Date: Mar 2009
Posts: 5,935
With some reasonable assumptions about all-in hands we can deduce that the actual frequency might be somewhat better than the 1/2538 for random hands. Players will tend to not hold 2's thru 5's as often as high cards when going all in. And although they will also tend to hold an Ace more often than random, that's just one rank vs the 4 ranks they will tend to not hold.
