Why Splitting Eights against a 10 in Blackjack is so Perplexing

10 October 2002

By Alan Krigman

To most folks, the ugliest possible blackjack scenario is a pair of eights against a dealer's 10-up. It's not merely that 16 is so weak in this situation. But Basic Strategy says to split the pair, and many solid citizens hesitate to put up twice as much money to buy two obvious underdogs instead of one. Neophytes, and quite a few veterans, view this as a dilemma and have been known to blaspheme by actually questioning "the book" on this hand.

Standing on 16 against 10-up, players win if the dealer busts and lose otherwise. Chance of winning is close to 23 percent; that of losing is the complement, just above 77 percent. Since the win or loss is one unit, expectation - which anticipates average result after numerous hands - is to lose a bit more than $0.77 - $0.23 or $0.54 per dollar wagered. Standard deviation, a statistical term characterizing the volatility of this bet, is 0.84 units.

Hitting, likelihoods are about 20 percent of winning, 74 percent of losing, and 6 percent of pushing. Compared to standing, chances are roughly 3 percent less both of winning and losing, the difference going to pushes. Expectation is to lose just under $0.74 - $0.20 or $0.54 per dollar bet. Standard deviation is 0.80 units, pushes being the reason for reduced bankroll volatility.

Splitting muddies the waters because players can not only push but win or lose multiple units. The accompanying analysis shows the amounts and associated probabilities, assuming players may only split once and will double on subsequent two-card 11s.

Chance of winning or losing up to
four units, splitting eights once only against
a 10 and doubling on subsequent two-card 11s

 

To determine expectation, add the products of the units won times their corresponding probabilities, then subtract the products of the units lost times their matching probabilities. The arithmetic yields just under $0.49 per dollar wagered at the beginning of the round. The standard deviation is 1.7 units, indicating that bankroll volatility is over twice that of either alternative.

Basic Strategy is predicated on the decision having the best - in this instance the least negative - expectation. Splitting beats hitting and standing by roughly $0.05 on the dollar. So this is what all the experts and the wallet-sized charts say to do. If expectation were the be-all and end-all of gambling, the case would be closed and everyone should go (or stay) home.

But, wait, there's more! Expectation is fine for projecting long term performance. Volatility tends to rule the short run - individual sessions, casino visits, most players' lifetimes. The analysis predicts losses in 53.4 percent of all splits, much less than the 77 or 74 percent for standing or hitting, respectively. The predominant loss is two units, at 37.7 percent. Win rate is 27.6 percent, somewhat better than the 23 or 20 percent for standing or hitting, respectively. The leading win, at 16.5 percent, is two units. Prospects of pushes are also high, 19.0 percent - up from zero for standing and 6.0 percent for hitting.

Expectation and therefore Basic Strategy favor splitting, to save a theoretical $0.05 on the dollar. Chances of winning and losing also show splitting to be the preferred option, with moderately greater numbers of wins, many more pushes, and far fewer losses.

Volatility, which has more overt impact than expectation or probability, is the bugaboo. This is most evident in two-unit losses, which occur more often than anything else and are widely considered the "usual" result, while the respectably sizeable chance of a two-unit win is normally seen as a "gift." So, many players sacrifice the nickel and hit rate, which are virtually invisible, to save the dollar they see and remember so vividly. The bettor's bard, Sumner A Ingmark, thought of it like this:

When I hear, "it's not the money," I keep trying,
To believe whoe'er was speaking wasn't lying.