6,7,8, frequency

[roliin]

How often do you get 6,7,8 in a six deck game, h17 ,das, ra, 75% pen? How often do you get it suited?

[Freightman]

How often do you get 6,7,8 in a six deck game, h17 ,das, ra, 75% pen? How often do you get it suited?

Just to clarify, you getting 678, or you getting 2 cards of the sequence with dealer getting the other

[DSchles]

Surely, you're getting a bonus for receiving 6,7,8 in your hand. But, there are other questions: Should the frequency simply be the probability of drawing those three cards, or, are we talking about getting a bonus and, therefore, having to decide whether it is worth drawing a third card, when it is otherwise the wrong basic strategy to do so? In that case, the ultimate probability of obtaining the hand is smaller than the aforementioned probability, because, well, there are some two-card combinations vs. certain dealer upcards that aren't worth hitting in an effort to obtain the bonus.

I discuss all of this on pp. 30-31 of BJA3. But, of course, the straightforward calculation for any 6,7,8 three-card hand, in 6-deck (the rules don't matter), is:
(24/312)(24/311)(24/310)(6) = 0.002757, or 1 in 363. Suited would be (6/312)(6/311)(6/310)(4)(6) = 0.0001723, or 1 in 5,804.

And, of course, to not double count, if you want to distinguish, you would have to remove all the suited 6,7,8s from the first group, so as not to double count them.

Don