CSM's

Does anyone know what the (approximate) mathematical potential is for a card from one game against a CSM to show up in the very next game that is dealt?

The probability for each card, for each hand, assuming a random shuffle, is identical. For a single-deck game, the first probability that the first card is any particular card is 1/52. The next card dealt's probability that it will be one of the remaining cards is 1/51; the probability that the second card will be the first card is 0. Etc.

For multi-deck games, it depends on whether you mean physically the same card, or a card of the same suit and value. If the former, 1 / (52 * num decks); if the latter, 1 / 52, same as above.

Probabilities are not influenced by previous hands/plays -- look up the "gambler's fallacy."

Hello Birdbeast:

Since I do not know exactly what goes on inside a CSM, the following ideas are only a guess. If a minimum of 6 cards are always kept in the dealing chute, and since they use 4 decks of cards, that means that the cards from the previous game (lets assume there were 20 cards used), are mixed together with 182 cards that are also inside the machine in a mixing area. This particular scenario would mean that the odds of selecting a card from the discards is 20/202 or approx. 10%. These odds would apply to the first card from the mixing area that is sent to the dealing chute.

I don't know the exact answer either but in order not to run out of cards during the initial deal (with fast dealers) the outlet buffer should contain something like 15 cards.
Even if the cards are reinserted after each deal which is rather an exception than the rule, the chances that one of these same cards will reappear on the next deal are very small, since they have to make it first to the wheel and then to the outlet buffer.
What I mean is: You don't have to withdraw your bet if you have detected an advantage situation and the dealer suddenly reinserts the discards before dealing.

Francis Salmon

Agreed, completely.

By "same card," I wasn't sure whether the original poster meant "same" RE: 6 of clubs is the same a 6 of clubs from another deck in the shoe, or "same" RE: a particular 6 of clubs (let's say, not that this would ever happen, with a particular coffee stain in a particular place) coming up in the very next deal.

Sorry for the confusion
-bb