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Best of Alan Krigman
Playing it smart - A betting strategy can't bring you luck9 April 2007
How does betting $12 on the six differ from dropping $6 each on the six and eight at craps? Either way, you lose $12 if the dice show a seven, a fate you figure will befall on the average of six times in every 36 throws. With the whole amount on the six, you expect to pick up $14 on the average of five times in every 36 throws. With the money split across the six and eight, you expect to be paid $7 on the average of 10 times in every 36 throws.
So, in 36 statistically correct trials, you'd lose $12 x 6 = $72 following either approach. The first way you'd win $14 x 5 = $70 and the second you'd pick up $7 x 10 = $70. The deficit a result of the house's edge on the bets is the same, $2.
Were edge the sum and substance of gambling, the two strategies would be equivalent. And, to the bosses, they are. This, because casinos have a long-term perspective in which the law of averages serves as a reliable indicator of performance.
To solid citizens, however, the alternate bets lead to sessions with distinctive characteristics. You can anticipate the contrasts intuitively by picturing fewer but larger hits in the one case and more but smaller returns in the other.
Continuing to reason intuitively, consider the implications of a little good or bad luck with either strategy. For the same shifts in hits or misses relative to the statistical projections, $12 on one number will cause your bankroll to rise or fall more rapidly than it would splitting across two. The former strategy should therefore give you a better chance at greater earnings but leave you more vulnerable to a wipe-out; the latter is apt to leave you less well off when fortune smiles, but simultaneously less likely to send you to the lockers when it frowns.
The effect isn't just some vague notion dependent on anything as ephemeral as luck. It can be analyzed rigorously by considering the volatility of the bets as well as the edge. Volatility, expressed by what the math mavens call "standard deviation," is a measure of bankroll fluctuations during single or multiple coups.
In 36 theoretical throws with $12 on the six, bankrolls should rise by $14 five times, fall by $12 six times, and not change 25 times. Standard deviation considering wins, losses, and no-actions works out to $7.16 per throw. With $6 on both six and eight, bankrolls should climb $7 10 times, drop $12 six times, and not change 20 times. Smaller wins, despite greater frequency, lead to lower standard deviation; it's $6.13 per throw.
Session profiles can be found by combining these standard deviations and the values of edge. Of particular interest are the likelihood of reaching a stated win goal before tumbling to a specified loss limit, and the prospect of remaining in contention above the same loss limit for a desired number of rounds.
Make believe a player has a $200 stake and is determined to double it or go broke trying. Intuition about greater bankroll swings suggests that the chance of success is better with $12 on the six than $6 each on the six and eight. The math not only confirms but quantifies this. The probability is 39 percent with the former and 36 percent with the latter.
For the same $200 budget, pretend the player wants to stay in the game for a minimum of 500 throws. Intuition insinuates that the smaller swings should make the chance of not dipping down to the bust-out point in such a session better with the split than the consolidated bet. Again, the math shows this is correct and also puts numbers on it. The probabilities are 74 percent with the single wager and 81 percent with the double.
This discussion has been framed in terms of a narrow craps example. Presumably, you've guessed that it applies broadly to situations in which you can elect to bet on propositions with low probability of high payoffs or the converse. For, as that giant of generalization the popular poet Sumner A Ingmark penned:
Bereft specific information,
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