Stud8 probability question

[anatta]

How do you figure this out?...

You start out with 3 to a low. You are playing 8 handed. You see 4 low cards up on third street, not your rank, that is, you can catch good with 16 cards on 4th. Its pretty easy for me to figure out that its 16/42 or 38.1% that you will catch good.

My question is what does the math formula and figures look like in trying to answer the question, "how often will you make your low by 7th". You have to hit twice and you have 4 chances.

[AaronBrown]

Getting an exact answer is a bit tricky and it depends on whether any of the cards you see are paired with each other. But you could get an approximate answer more quickly by ignoring some of the details.

You will get 4 cards out of the 42, which can happen C(42,4) = 42*41*40*39/(4*3*2*1) = 111,930 ways. You need to count how many of these give you an 8 low or better.

It's tempting to say there are 16 cards you could get for first low card, 12 you could get for the second (since the second can't pair the first), then any of 40 remaining cards for the third card and any of 39 remaining cards for the fourth. You have to divide this by 4 because the first two or last two cards could come in either order so it's 16*12*40*39/4 = 74,880. 74,880/111,930 = 0.67 or about 2 chances in 3.

Unfortunately, it's trickier than this. It's true there are 16 good cards for the first card, but there could be anywhere from 12 to 15 for the second, depending on whether the first card paired one or more of the other players' upcards. Then the trouble with saying the third card can be any of 40 is that you will double count cases in which you get three or four good low cards.

It's not really hard to do this problem, but you have to add up a lot of cases.

[anatta]

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