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# How pretty are blackjack players sitting with totals of 20 or 21?

9 November 2009

Most blackjack buffs believe they're in the catbird seat with totals of 20 or 21. Yes, these hands can lose to blackjacks. But solid citizens learn this particular bad news before the round starts to be played out. So, if the dealer shows a 10 or ace and the hole card check proves benign, or the upcard is anything from two through nine, the payoff is practically headed to the bank.

Right? Maybe. It depends on your definition of "practically." And it also depends, in a way that might surprise you, on the upcard.

For the game taken as a whole, considering both the value and the prospects for the occurrence of the various upcards, players with totals of 20 average 70.0 percent wins and 18.0 percent pushes. Losses to blackjacks come in at 4.7 percent and to other totals of 21 at 7.3 percent – 12 percent in all.

Bettors holding non-blackjack 21s are even stronger. They average 88.0 percent wins, 7.3 percent pushes, and 4.7 percent losses.

Knowing the overall probabilities of winning, pushing, and losing with these hands may be intellectually pleasing. So, if you run into any intellectuals at the blackjack table, fill them in.

More critical to bettors in the heat of the action is the chance of winning, or at least pushing, but not losing, against whatever upcards are in front of them. This information helps people cope with the frustrations arising when what they think is a mighty hand goes amiss. Folks who might otherwise worry whether there's something about the dealer, the cards, their mismatched socks, or even fate itself that's conspiring against them.

The accompanying table shows probabilities of alternate outcomes with totals of 21 against each dealer upcard. Data are for 8-deck games where the dealer stands on soft 17.

Probabilities of winning, pushing, or losing with totals of 20 or 21 against various dealer upcards in an 8-deck game

```	player has 20	player has 21
upcard	win	push	lose	lose	win	push	lose
(bj)	(other)			(bj)
2	75.8%	12.4%		11.8%	88.2%	11.8%
3	76.4%	12.1%		11.5%	88.5%	11.5%
4	77.2%	11.6%		11.2%	88.8%	11.2%
5	78.0%	11.2%		10.8%	89.2%	10.8%
6	80.1%	10.2%		9.7%	90.3%	9.7%
7	84.7%	7.9%		7.4%	92.6%	7.4%
8	86.1%	6.9%		6.9%	93.1%	6.9%
9	81.9%	12.0%		6.1%	93.9%	6.1%
10	54.8%	34.1%	7.7%	3.5%	88.8%	3.5%	7.7%
A	50.7%	13.1%	30.8%	5.4%	63.8%	5.4%	30.8%
overall	70.0%	18.0%	4.7%	7.3%	88.0%	7.3%	4.7%

```

Your intuition will undoubtedly suggest that hands of 20 are especially likely to push when the dealer shows a 10. This, simply because 10s are fourfold more frequent than other ranks and their appearance as hole cards leads to a push. This is borne out by the data in the table. The high frequency of pushes contributes to the middling expected instances of wins and the low rate of losses. Wins against aces are also only moderate – here primarily because those 10s in the hole make blackjacks more common than any specific alternate result.

The other upcards yield the surprises. In general, in the normal course of a game, six-up makes dealers most susceptible to being beaten by players, owing to a combination of busts and low finishing totals. The vulnerability decreases as the exposed rank drops to two, then continues to decline from seven through ace.

On a player's 20, the shot of winning is least against a two – 75.8 percent, and greatest versus an eight – 86.1 percent. The chance of losing works oppositely; it's highest facing a two – 11.8 percent and lowest looking at a 10 – 3.5 percent. Pushes form the difference. A 21 can't lose except to a dealer's blackjack. But its hopes of winning rise from 88.2 percent opposing a two to 93.9 percent against a nine. Non-winners push.

Not that, in any absolute sense, a 75 to 86 percent chance in the case of a 20, or an 88 to 94 percent shot with a 21 is shabby. But those who mistake these figures for certainty of success might well ponder the pertinence of this poignant poetry by that petulant pencilpusher, Sumner A Ingmark:

The hands they think are tough to vanquish, Cause optimistic gamblers anguish.

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Best of Alan Krigman
Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.