How other Players at a Blackjack Table Affect Your Game

Enlightened blackjack enthusiasts seem willing to accept what the gurus assert about hands on which to hit, stand, double down, split, and surrender. Usually on faith, without questioning the math behind Basic Strategy. This, despite the counterintuitive nature of edicts such as "always split eights, even against 10s" or "hit 12s against twos." But many of the same solid citizens reject, or at least seriously doubt, the experts' contention that their chances aren't affected by how others at the table play.

Skeptics point to cases when a novice who "shoulda stood" instead "took my 10" or who "shoulda hit" didn't and "fed the dealer a five," and by doing so "ruined the game for everyone else." Not many remember when it went in the opposite direction. And, those who do recall such circumstances, consider them the rare exceptions rather than the rule. Further, most people confuse the bearing of an event on the probability of a result before the fact, with its influence on the outcome after the dust settles.

Consider the following. It's you, and one other player sitting on your right. Your neighbor, who goes first, has 10-5. You, A-7. The dealer gets 10-up. By all that's holy, you both should hit. You're underdogs regardless, less so by hitting than standing.

You're praying for a two or a three, to sneak in with a total of 20 or 21 and gain the advantage. What are your chances? To get an answer, assume an eight-deck game in which you know only the five cards you can see at the moment. This leaves 416 - 5 or 411 "unknown" cards, including 64 twos and threes.

Make believe you're sitting with a neophyte, who stands. The chance you'll receive a two or three is 64/411 or 15.57 percent.

Instead, say you're with a seasoned veteran, who hits. The chance he or she will get a two or three is 64/411 or 15.57 percent. The outlook for anything else is the complementary 84.43 percent.

After your cohort hits, 410 cards remain. If he or she gets a two or three, one less is left for you; your chance of pulling either of these is then 63/410 or 15.36 percent. Conversely, if he or she draws a different rank, your prospect of receiving a two or three is 64/410 or 15.61 percent. Overall, the chance you'll get a two or three is (0.1557 x 0.1536) + (0.8443 x 0.1561). This equals 15.57 percent, as it was when the other player stood.

This example shows that your companion's decision changes the structure but not the value of the expectation for your hand. Similar analyses yield identical conclusions for any conditions. So, unless you think occult forces order the cards in a shuffled shoe, previous players' actions don't affect your chances.

In two ways, however, other people at the table do influence your game. Here's how.

First, casino shuffles don't truly randomize a deck or shoe. The closer that cards are to one another before the shuffle, the nearer they tend to be afterwards. More players at a table mean larger post-shuffle separations between the initial two cards received by each person, including the dealer. This increases the effective randomness of the round.

Second, the more spots in action at a table, the fewer hands dealt to each position per unit time. Most bettors don't notice this factor. An hour of action seems no different playing one spot, one-on-one with the dealer, and receiving about 200 hands, or if six individuals have one spot each and get only 60 hands. However, the number of hands per unit time has a major impact on the bankroll swings each participant is apt to undergo during any interval. In the course of two hours, one-on-one, someone betting $10 per round can be 95 percent confident of not peaking over $900 up or down. In two hours with six players at a table, the 95 percent confidence level applies to highs or lows of $500.

So, other players affect your game. But not in the manner you may have thought. A situation rendered in rhyme by Sumner A Ingmark:

When preconceived notions do swallow up fact,
Then folks on emotions not logic react.