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# Risk and Reward: You Can't Have It Both Ways

4 January 2000

Every casino game offers a trade-off between the chance you'll win and what you'll collect if you do. Players who understand gambling as well as they make believe, decide in advance whether they want a good shot at a small payday or prefer to buck long odds against rich rewards, then pick their wagers accordingly.

In most games, on a bet-by-bet basis, the probabilities and amounts are predetermined. For example, if you drop \$1.25 into a jacks-or-better video poker game and play properly, your chances are roughly 55 percent of a loss, 21 percent of a push with a high pair, 13 percent of a \$1.25 win with two pair, 7 percent of a \$2.50 win with three of a kind, and so on. Likewise, if you place the five or nine at craps for \$5, you have a 60 percent chance of a loss and a 40 percent chance of a \$7 win.

Roulette offers more choices than a chef at an all-you-can-eat buffet. Assume it's a double-zero game -- a 38-groove wheel. And to facilitate comparisons, say you'll always bet \$35 per spin.

At the high-probability low-payoff end, you can toss \$1 onto each of 35 numbers. If any one of them hits, you'll clear \$1 profit. The chance this will occur is 35 out of 38 or 92.1 percent.

At the low-probability high-payoff end, you can stack the entire \$35 on a single number. If it hits, you'll net \$1,225. The chance this will happen is one out of 38 or roughly 2.6 percent.

Of course, you can go between these extremes. For instance, \$5 on each of seven numbers gives you a seven out of 38 or 18.4 percent chance to win \$169. And \$7 on each of five numbers offers a five out of 38 or 13.2 percent chance to net \$241.

As far as the casino is concerned, these are all equivalent. You're putting \$35 on a bet where the house has a 5.263 percent edge. So the bosses figure they earn this fraction of \$35 -- just over \$1.84 -- every time you do it. They don't care whether their net comes from lots of little collections and a few not-quite-compensating big payouts, or the converse. This, because they average their action over many players and numerous rounds.

As an individual, however, your wagering strategy makes a difference. No one spin will ever cost you \$1.84. Instead, you'll lose \$35, or win either \$1 or \$1,225, depending on your approach.

What if you bet on two spins? You won't lose \$1.84 both times for a total of \$3.68. To envision what may actually happen, picture 1,000 solid citizens trying the two rounds. On the average, with \$1 on each of 35 numbers, 848 will win twice, earning \$2; 146 will prevail once and fail once, losing \$34; and six will be doubly drubbed, losing \$70. Similarly, with \$35 on a single number, the 1,000 stalwarts will fare as follows: one (well, not quite one, if you have to be pedantic) will win twice, grabbing \$2,450; 51 will win once and lose once, boarding the bus with \$1,190; and 948 will get zonked both times, losing \$70.

If you play a lot, your balance should start to close in on what the statistical whizzes in the bowels of the casinos predict. After 10,000 spins at \$35, with either bet, you should approach a \$18,400 loss. And after a million spins, you should be in the neighborhood of \$1.84 million in the hole. But, even here, your style of play matters because it determines just how far up or down from these conjectural depths you might really be.

To quantify the fluctuations, imagine betting the two ways on 10,000 spins. At \$1 on each of 35 numbers, your chance of being even or ahead is only 2.5 percent. The flip side is that you have 2.5 percent chance of being over two times \$18,400, or \$36,800, down. At \$35 on a single number, your chance of being even or showing a profit is improved -- to 18 percent. But you also have an 18 percent chance of being below \$36,800. Gambling not being a '48 Studebaker, you can't drive it to your advantage coming and going. Sumner A Ingmark, poet of paradox, may have said it best:

Casino bets, low risk but well endowed,
By laws of compensation, disallowed,
'Cause every silver lining has a cloud.

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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.