In this Blackjack Forum article, Arnold Snyder discusses the effects of a bad blackjack player on the win/loss results of other players at the table.
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The Bad Player at the Blackjack Table: Do Other Players Steal Your Aces?

 
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The Bad Player at the Blackjack Table: Are Your Results Affected by Other Players?

By Arnold Snyder
(From Casino Player, September 1993)
© 1993 Arnold Snyder

Question from a reader: I disagree with the common ďwisdomĒ that says that a blackjack playerís expectation is not affected by other players at the table. Iím not talking about poor players getting the cards ďout of orderĒ ó by taking the dealerís bust card, etc. I understand the arguments against that reasoning. Itís my contention, however, that even if you had all perfect basic strategy players at the table, the more players there are, the worse it is for all of them.

I was a dealer in Nevada some years back, so Iím speaking from experience. When I faced one player head-up, if that player followed basic strategy pretty closely, I would not win any sizable amount of money. If they were flat betting silver, they could play for hours with a $20 bankroll. Fill the table with players, however, and none of them could last for long. Iíd beat everybody.

There is a simple mathematical explanation for this. A blackjack occurs about once every 20.7 hands ó letís just say every 21 hands. If thereís just one player and the dealer, this limited number of blackjacks gets divided up between them 50-50.

As soon as another player enters the game, however, that player will get a share of the blackjacks. You're not splitting these opportunities 50-50 with the dealer anymore. Now youíre each getting one-third, and as you add even more players, each playerís percentage of blackjacks keeps going down. With 7 players, plus the dealer hand, youíll only get 1 out of 8 of all the blackjacks dealt. They get evenly distributed to every hand over the long run.

Itís almost impossible for any player to win at a full table, except by occasional luck. To put it another way that anyone can understand ó there are only a limited number of aces available (exactly 4 per deck!). When itís just you vs. the dealer, each of you will get half the aces over the long run. You will never get half the aces, for any lengthy period of time, at a full table. Itís just too many players fighting for too few possible aces. Iíve watched this with my own eyes.

No card counting system can change these facts. The only way you could ever beat the dealer is head-up. Why donít blackjack ďexpertsĒ print these facts? You should tell blackjack players the truth.

Answer: Frankly, I donít trust a lot of what many so-called ďexpertsĒ say myself. There are a lot of phonies out there, including a few of the most famous. Mistrust is very healthy in this field of expertise, and I donít want to discourage you from skepticism. On this point, however, you are wrong. Your argument sounds good, but the logic is flawed.

Letís get technical . . .

You are correct that approximately 1 out of 21 hands is a blackjack. Itís pretty easy to figure this out mathematically. Letís do it. First, a math lesson . . .

Suppose you got paid a bonus anytime you were dealt an ace as your first card. How often would you get the bonus? Thatís easy . . . With 4 aces in a 52-card deck, 1 out of every 13 cards is an ace. Youíll get your bonus on one out of every 13 hands.

Does it matter how many players are at the table?

No. Every player at the table will get the bonus on 1 out of every 13 hands in the long run. If itís just you and the dealer (2 hands total), then about once every 6 1/2 rounds either you or the dealer will be dealt an ace as a first card. You, personally, will get one every 13 hands. So will the dealer.

If there are 12 players (a giant-sized bj table!) plus the dealer (13 hands), then on average one hand per round would be expected to get a first card ace. You, personally, will still get a first card ace on only 1 out of every 13 rounds, but youíll see an average of one per round dealt to somebody.

If this makes sense to you, then Iíve just tricked you into understanding more about probability and statistics than you ever thought youíd learn. Because now Iím going to show you how to figure out that youíll be dealt a blackjack 1 out of every 21 hands, regardless of the number of players at the table.

There are only two ways to be dealt a blackjack. One: to be dealt an ace as a first card with a ten as your second card. Two: to be dealt a ten as your first card with an ace as your second card. Thatís it. There is no other way to be dealt a blackjack. If your first card is anything other than an ace or ten, forget it. Also, your second card must be a ten (for your ace), or an ace (for your ten). Thatís the sum total of all the possible ways of getting a blackjack. Hereís how we statisticians figure out how often this will occur (somewhat simplified):

1 out of every 13 first cards will be an ace.

How many of these will end up as blackjacks?

4 out of 13, since there are 4 ten-valued cards (10, J, Q & K) out of the 13 possible card denominations.

Hereís the hard part: This means that 4/13 of 1/13 of my hands will be a blackjack with an ace as the first card. What is 4/13 of 1/13? If you do the math, itís 4/169, or approximately 1 out of 42.

So, 1 out of 42 hands will be a blackjack with a first card as an ace.

The only other way I can get a blackjack is with a first card of ten. How often does this occur? If you do the math here, youíll come up with the same answer. Instead of 4/13 of 1/13, we take 1/13 of 4/13, and the answer is identical: 4/169, or about 1/42.

So, on 1 out of every 42 hands, youíll get an ace-first blackjack; and 1 out of every 42 hands youíll get a ten-first blackjack. This means youíll get a blackjack on 2 out of every 42 hands, or 1 out of every 21.

It makes no difference how many players are at the table. You are making the mistake of believing that youíll be receiving fewer than 1 out of 13 aces as more players join the table. Although the percentage of the total blackjacks dealt that you personally receive does go down as more players join the game, the percentage of blackjacks occurring per round goes up. Your blackjack expectation remains exactly the same.

You can test this fairly easily at home. Just deal 2-card hands. You donít have to play them out. Keep track of the total # of hands dealt and the total # of blackjacks. Within a couple thousand hands youíll see that itís approximately 1 out of 21. Whether you deal these hands out 2 at a time, 3 at a time, or 5 at a time, it makes no difference. About 1 out of 21 hands will get a blackjack. Try it!  ♠

For more information on every aspect of the game of blackjack, see Arnold Snyder's Big Book of Blackjack. For information on how professional gamblers beat the game of blackjack, see Blackbelt in Blackjack by Arnold Snyder.

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