In your first scenario, where the dealer fails to burn a card before the flop, and before action takes place, I would NOT use the next card as a four card flop situatuion. The next card, or the fourth card off the top, would have been one of the proper flop cards! This we know for sure, right? Consider the flop on the table has one card that should have been the burn. That's bad enough. To take the next card and scramble it into the incorrect flop will only compound the error and expose yet another card!
Thanks, Nick. First, with respect, I would never call a misdeal. I think it is too drastic of an action in general, given that most of the proper flop is still preserved. We should always be striving to play hands to completion, particularly since there has already been betting action in the hand; if I had to choose between a misdeal or a reshuffling of the flop, I would choose the latter. Thankfully, those are not the only alternatives.
However, I do agree that it actually makes more intuitive sense that the fourth card (which we would know would have been part of the flop) be used, and the current flop scrambled to remove one as the burn (again assuming we didn't know which one the proper burn was to fix it). At least then we have a 1/3 chance of actually having the flop that was supposed to come out.
I guess as a practical matter though, there isn't a basis for this in the current rules. RROP says we must reshuffle, and TDA only provides for a four-card flop shuffle. So of the two, I think the four-card approach is preferable, if we are forced to choose.
That being said, I don't actually think simply using the four-card approach in this situation is that bad. By taking the next card and scrambling it into the original flop, we aren't really "compounding the error" as you might suggest because the fourth card that might be exposed is the same card that you would have exposed and kept on the new flop had you performed the three card scramble anyways. In fact, I believe that if you work out the math, 75% of the time when using the four-card scramble, you will get the same result as if you had used a three-card scramble.
For example: Suppose A, B, C are on the flop, D is on the deck. The flop should have been B, C, D with A as the burn.
1. Three-card scramble - keeping D on the new flop
- 1/3 of the time, new flop will be A, B, D
- 1/3 of the time, new flop will be A, C, D
- 1/3 of the time, new flop will be B, C, D
2. Four-card scramble - mix D with old flop
- 1/4 of the time, new flop will be A, B, D
- 1/4 of the time, new flop will be A, C, D
- 1/4 of the time, new flop will be B, C, D
- 1/4 of the time, new flop will be A, B, C
If you compare, you will see that that two approaches largely give the same results, except that in the four-card scramble, you may get the original flop back 1 in 4 times. I don't think that is a huge deal though, since it happens relatively infrequently; the good thing is that D will still remain hidden, no new cards have been exposed, and all players are at an equal disadvantage.
In summary, I think the four-card scramble is an acceptable alternative to the three-card scramble, allows us to point directly to an existing TDA rule when we use it, and as Stuart points out earlier in the post, it is just that much simpler to implement in practice.