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# How good is your chance of a run of modest profits?

25 October 2010

Erudite gamblers know that prospects for successful sessions are greater, the more modest the earnings peaks at which they'll quit relative to the depths of the holes they're willing to dig trying. By setting win goals below loss limits, they can get a stronger shot at victory than defeat.

To pin some figures onto those chances, say you're a blackjack buff who follows precise Basic Strategy, holding the house to 0.4 percent edge. Suppose you start with a \$200 stake. Betting \$5 per round, the probability you'll win \$50 before losing \$200 is 77.4 percent. Betting \$10, your prospects rise to 78.7 percent. If your objective is to win \$100 before losing \$200, the probabilities fall to 62.4 and 64.6 percent wagering \$5 and \$10, respectively, per round.

House advantage, of course, is a factor. With no edge, regardless of bet size, the chance of winning \$50 before losing \$200 would be 80.0 percent and that of winning \$100 before losing \$200 would be 66.7 percent. In the other direction, at blackjack, a solid citizen who doesn't always hit or stand "by the book," sometimes buys insurance, and skips marginal splits and doubles might give the house 1 percent edge. At this level, the likelihood of reaching \$50 before losing \$200 is 73.3 percent betting \$5 and 76.8 percent at \$10. With 1 percent edge, for a \$100 win goal, chances of victory are 55.9 percent at \$5 and 61.3 percent at \$10.

These values might suggest you're onto a "money management" secret that'll let you visit your friendly, neighborhood casino to collect your day's pay as often as you wish. Chance of success around 75 percent (3-to-1) needs no apology. Further, this win-to-lose potential holds regardless of past outcomes. But, how apt are you to have a string of winning sessions, each with prospects of joy in this range? Consider the best of the cases described, 78.7 percent chance of winning \$50 before losing \$200 by following perfect Basic Strategy and betting \$10 per round. Two consecutive triumphs come in at 69.1 percent, three fall below half at 48.7 percent, and so on; by 10 in a row, you're down to 9.1 percent.

Playing five sessions and winning them all, a \$250 gain, is a 30.2 percent shot under the cited conditions. Winning four out of five breaks even – four \$50 wins and one \$200 loss – with 40.8 percent chance. Fewer successes are net losses, from \$1,000 winning none to \$250 scoring three.

Five sessions are too few on which to test a gambling hypothesis. Consider, instead, 54 sessions – still not many in the grand scheme of the universe, but enough to show the effect. Win 44 or more and you're up between \$200 and \$2,700. The chance of ending in this range is 38.1 percent.

The 54 sessions are independent trials. So, given 78.7 percent chance of winning one, you expect to average 0.787 x 54 or 42.5 successes and 11.5 failures. You'd theoretically win \$50 x 42.5 or \$2,125 and lose \$200 x 11.5 or \$2,300, putting you \$2,300 - \$2125 or \$175 behind. This is what the house makes from edge. Since you bet \$10 per round at 0.4 percent edge, every wager accounts for \$0.04. The average number of rounds to reach a quitting point in the 54 idealized sessions is accordingly \$175/\$0.04 or 4,375, which reduces to 4375/54 or 81 rounds each.

At \$5 rather than \$10 per round, the probability of being ahead by winning 44 or more of the 54 sessions falls from 38.1 to 29.7 percent. The average number you expect to win is 0.774 x 54 or 41.8. At this level, you'd theoretically have taken in \$2,090 and given out \$2440 for a net loss due to edge of \$351. Betting \$5 at 0.4 percent, edge gives the house \$0.02 per round. This means you'd have played \$351/\$0.02 or 17,550 rounds – an average of 17,550/54 or 325 per session.

Dropping from \$10 to \$5 bets, you get to play four times as many hands. However, the house earns twice as much from edge – a result reflected in the likelihood you'll reach your goal in 31.8 - 29.7 or 8.4 percent fewer sessions. Either way, the analysis shows that tactics favored to win a session, without regard to the sizes of payoffs and losses, don't circumvent the advantage the casinos build into the game. The clever casino Calliope, Sumner A Ingmark, captured the crux of this concept when he crayoned:

Succeeding once may not be tough, but don't let winning fool you,
It's in the long term things get rough, when laws of nature rule you.

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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.