The dealer got six 10 cards first up in a row. The odds of this are a 1,178-1.
by: RobbieT
This number is not correct. In a six-deck shoe, there are 312 cards, of which 96 cards are Tens. In the 1st round of playing, the probability of a Ten as the dealer upcard is
96/312.
In the 2nd round, the probability of a Ten as the dealer upcard is
(95-5x95/311)/(312-6),
where 5x95/311 is the number of Tens that are used in the five other cards in that round. Here we assume there is only one player at the table and each round uses six cards on average.
In the 3rd round, if we set t= 95-5x95/311, then the probability becomes
[(t-1)-5x(t-1)/(311-6)]/(312-12).
In the 4th, 5th, and 6th rounds, the probabilities have the same recursive formula.
Therefore, the probability of six Ten-upcards in a row is the multiplication of all these six terms, that is
0.000758, or 1 in 1320
*** (I've thought about this again. It looks like Don is correct, so I made revisions. No simulation is needed here.)