What Are the Odds?
by Arnold Snyder
(From Casino Player, July 1995)
© Arnold Snyder 1995
Question from a Blackjack Player: What are the odds against three players and the dealer all getting a blackjack on the same round? This actually happened to me on my last trip to Las Vegas. The dealer was so amazed, she called the pit boss over to see it, and he said he’d never seen anything like it before. As a side note, I was the only player who took even money when the dealer asked us if we wanted to take insurance. (She had the ace up.) Both the other players declined because they thought it would be impossible for the dealer to also have a blackjack. I figured if the dealer had just dealt three naturals, she most likely dealt a fourth! I was right! Have you ever heard of anything like this?
Answer: There are a couple of pieces of information lacking from your description of this event that are crucial to analyzing the exact probability of occurrence. First, you don’t say specifically whether or not any other players, who did not receive blackjacks, were playing hands at the same table, or if any of the three of you who received blackjacks were playing more than one hand (one of which was not a blackjack). I’m going to assume that only three players were at the table and that each player was playing only one hand.
Obviously, if four out of seven or eight hands dealt were blackjacks, it would be far less unusual than if four out of four hands dealt were blackjacks. (Or, at least, this would be obvious to anyone who’d taken an introductory course in probability and statistics. It may not be obvious to you, but take my word for it.) I’ll make the assumption that there were only four total hands in play — three players and the dealer — because you relate that the dealer was “amazed” and the pit boss stated he had “never seen anything like it before.” I’m sure most dealers and pit bosses of any experience have seen four simultaneous blackjacks dealt at a full table of players, in which three or four non-blackjack hands were also dealt, rare as even this would be.
The other pertinent fact you fail to mention is how many decks were in play. This is a crucial detail if you want to figure out the precise likelihood of occurrence. In a single-deck game, where there are only four aces in play, it would be far less likely for one of each of these aces to be dealt to each of four players than it would be in an eight-deck game where four aces represent only 12.5% of the total number of available aces in the shoe.
Technically, this is a fairly simple blackjack math problem to figure out, and you can easily do it on any pocket calculator, though it is a bit tedious. You simply calculate how many total possible ways four simultaneous blackjacks can be dealt, out of all the possible four-hand two-card combinations, and you get the odds against it occurring. Of course, it could take you a month of Sundays if you’re going to sit there and actually run through every possible two-card combination, four hands at a time, then count the totals of those which are four blackjacks vs. those which are not; and to do this for an eight-deck game, would take you multiple lifetimes. Fortunately, there are easy shortcut methods for figuring out problems like these.
In a single-deck game, the odds against being dealt four blackjacks out of four hands are about 1.8 million to 1. Most of us are unlikely to ever see such an event. A dealer who deals only single-deck games, 40 hours per week, always to three players at a time, at the average rate of 400 hands per hour, would likely see this about once every 112 weeks. Since dealers actually have constantly varying numbers of players when they deal, it’s probable that many full-time dealers would not actually experience such an occurrence as you witnessed in their careers.
In an eight-deck game, the odds against this are only about 237,000 to 1. For any number of decks between one and eight, the odds against this occurring will fall somewhere between these extremes, the fewer the number of decks in play, the greater the odds against it occurring. In any case, it’s not something that happens frequently by any means.
Regardless of the number of decks in play, if you were not counting cards, and you had not been keeping track of the ratio of tens to non-tens which had been dealt at the time these four simultaneous blackjacks had been dealt (and I assume you had not), you made a mistake when you took insurance. I realize you were the only player who made money on the hand, and that you “can’t lose” whenever you insure a natural, but the fact is, purely from the perspective of the statistician, the odds were strongly against the dealer having a ten in the hole. So, remember that the next time this happens to you! Don’t fall for that sucker insurance bet. (For some reason I feel you’re not going to take my advice on this one. . . .) ♠