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Bad Blackjack Players Don't Hurt Your Odds

11 January 2004

By Fred Renzey

Throughout the annals of established blackjack literature, you've read that a bad player at 3rd base can't hurt your own chances in the game. In public seminars, I've presented illustrations with cards on sequential billboards that showed that same effect -- or lack of. I've even reported the results of my own 500-hand experiment that showed that 1st base did no worse when 3rd misplayed all 500 of his hands as when 3rd Base played perfect basic strategy.

Still, the myth persists. Repeatedly, I get e-mails, letters, complaints and remarks from other players who insist that playing with "ploppies" (bad players) is the reason why good players lose. So there's only one thing left. I hesitate to do it because I know how much most of you hate numbers, but you've forced me into it. I have no choice but to present you with the simple mathematical proof that another player's mistake has absolutely no effect on your own chances to win your hand.

Now don't put words in my mouth. I didn't say that another player's mistake couldn't cost you your hand. Indeed it could. It could also win your hand for you. It works both ways. And because it works both ways randomly, it'll all wash out in the end. The real point is that another player's mistakes don't hurt your own odds to win your hand.

Here's an exemplary situation. I've simplified it to make things as clear as possible. You're at 1st base and have 14. Third base also has 14. The dealer's got a 5 up. You've already stayed pat and now it's 3rd base's turn. As the dealer points to the 3rd baseman, she accidentally snags her hole card on her sleeve and flips it face up. It's a 10! So now we know she positively has 15.

Suppose we also just happen to know that there are only five cards left in the shoe and they consist of four 10s and one 6. So if 3rd base hits, he has four chances out of five to bust with a 10. But if he stands, the dealer has four chances out of five to bust. Obviously then, 3rd base's best chance to win his hand is by standing.

The real question however is, which 3rd base play gives you at 1st base the best chance to win your own hand? This is where very few people can see the forest through the trees. So let's remove those trees, one by one.

Since there are five cards left, we'll theoretically play the hand out both ways five times each. If 3rd base stands five times, the dealer will (on average) bust with a 10 four times and catch the 6 to make 21 once. Therefore, you over at first base will win your hand four out of five times. That's any 80% win rate.

Now if 3rd base hits five times, he'll needlessly bust himself with a 10 four times. That's why hitting is such a bad play for him -- yet not for you who have already stood! You see, those four times when 3rd base catches a 10, there will be three 10s and one 6 left for the dealer. So the dealer will catch a 10 three out of those four times. And that fifth time when 3rd base caught the 6, the dealer will positively get a 10. So when 3rd base hits then you at 1st base still win your hand four out of five times -- the same 80%. 3rd base's bad play has hurt his own chances, but not yours!

That's the long and short of it. Now for those of you who are adept at 8th grade math, here's the mathematical proof I promised you -- in layman's terms.

When 3rd base stands:
Dealer's probability for a 10 is 80%.

If 3rd base hits:
3rd base's probability for a 10 is 80%.
That 80% of the time, the dealer's probability for a 10 is 75%.
80% x 75% = 60%.
3rd base's probability for a 6 is 20%.
That 20% of the time, the dealer's probability for a 10 is 100%.
20% x 100% = 20%.
The dealer's total probability for a 10 is 60% + 20% = 80%.

Fred Renzey
Fred Renzey is a high-stakes, expert poker player. On a daily basis he faces--and beats--some of the best players in the country in fierce poker room competition. Now for the first time, Renzey offers his perceptive insights on how to play winning poker. For Fred's 13-page blackjack booklet "Ace/10 Front Count", send $9 to Fred Renzey, P.O. Box 598, Elk Grove Village, IL, 60009

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