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# Can you afford to get an edge at blackjack by counting cards?

17 May 2010

Casino bets are as tough to compare with one another as apples and oranges. House advantage or edge is often used to gauge alternatives. However, this criterion is really of value only to the bosses because its relevance lies in calculating long-term averages. Edge says little about the short-run bankroll swings solid citizens experience during one or a few single sessions or casino visits, absolutely or in correlating different wagers.

Even-money equivalents offer an intuitively satisfactory yardstick to measure alternative wagers. These hypothetical propositions have two parts, amounts at risk and corresponding chances of 1-to-1 payoffs having the same edge and volatility as actual bets. For the mathematically minded, these statistical components are calculated as follows: a) Amount equals the actual average bet times the square root of (the sum of the variance plus the edge squared). b) Chance of winning is 0.5 plus (half the edge – negative if it favors the house – divided by (the square root of the sum of the variance plus the edge squared)).

The utility of even-money equivalents can be illustrated in terms of a Blackjack buff thinking about advancing from Basic Strategy to card counting. The essence of card counting is that players have an edge when more low than high ranks are gone from a shoe so prospects have become above average of drawing nines, 10s, or aces. Systematically profiting from this information requires raising bets as conditions improve. But, chances of winning particular hands don't necessarily increase enough to keep high wagers from exposing players to bankroll-busting downswings.

The even-money equivalent of \$25 blackjack bets in every round following Basic Strategy is a \$28.17 wager with 49.78 percent chance of winning and the complementary 50.22 percent chance of losing. The excess over \$25 accounts for splits and doubles as well as 3-to-2 black-jacks. The 0.44 percent negative offset between chances of losing and winning represents the house edge.

To obtain corresponding figures for card counting, suppose that a player achieves a 1 percent edge over the house by betting the amounts indicated, at the frequencies shown, in the accompa-nying somewhat contrived table. The data would presumably be based on the likelihood of counts reaching the appropriate betting levels.

Representative amounts and frequency of bets
for an aggressive card counting strategy

```	amount	frequency
\$10	51.08%
\$25	27.78%
\$50	12.28%
\$75	8.19%
\$100	0.67%
```

The average bet for this card counter is \$25. The even-money equivalent is a flat wager of \$162.70 per round, having 50.08 percent chance of winning and 49.92 percent chance of losing.

The whopping increase from \$28.17 to \$162.70 for the same \$25 average pre-round wager results mainly from the wide spread of bets made by the counter. And, although a situation may be so promising as to warrant a \$100 bet, it's no guarantee of a winning hand. Further, the counter's net favorable 0.06 percent probability margin is numerically smaller than the adverse difference in chances for the Basic Strategy player.

Over, say, 100,000 hands, the \$25 Basic Strategy bettor would have a gross wager of \$2,500,000 and expected loss of \$12,500. The card counter under the stated circumstances would have the same handle and expected profits of \$25,000. But the even-money equivalents of \$162 versus \$28 suggest that the counter would need a considerably larger stake than the Basic Strategy player to outride the normal downswings of typical sessions. This, along with the mettle to push out the big bucks when the probability of a winning hand goes above 50-50, but not by much. Which explains why urban myths about card counting success tend to involve investor-backed teams and not individual players. Here's how the virtuoso of verse, Sumner A Ingmark, depicted the dilemma:

The betting schemes you live in fear of,
Despite successes you may hear of,
Are strategies you should steer clear of.