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# Is a big win on longshots guaranteed if you make enough bets?

15 February 2010

Unlike lotteries, ball games, or horse races, casino gambles typically involve repetitive bets. A previous article discussed the chances of winning or losing various amounts, using alternate strategies for series of wagers on even-money propositions.

Many solid citizens, however, avoid even-money games. They favor risking what they view as chump change on longshots touting rich rewards. How are chances and corresponding net gains or losses enhanced betting in multiple low-probability high-payoff coups?

Pretend you drop \$5 onto a single spot at a double-zero roulette table. The probability is one out of 38, or 2.63 percent, you'll earn 35 x \$5 or \$175 and the complementary 97.37 percent you'll lose your nickel. Instead, say you start with \$100 and bet \$5 on one spot in separate rounds until you score or go broke. The chance you'll win on the first spin and grab \$175 remains 2.63 percent. Losing on the first and winning on the second, a likelihood of 2.56 percent, nets you \$170. Or, envisioning a two bet series, you have a 2.63 + 2.56 or 5.19 percent shot to win \$170 or \$175. If you're willing to venture \$5 up to 20 times, quitting as soon as you hit, prospects are 41.34 percent you'll win from \$80 to \$175, versus 58.66 percent you'll lose \$100.

You can have over 50 percent chance of a win – in the \$50 to \$175 range – starting with \$130 for as many as 26 spins. Another option would be to begin with \$180, sufficient for 36 spins. Now, a win on the first is still worth \$175. A hit on the 35th yields \$5 – the \$175 you pick up minus the \$170 lost on the 34 misses. Succeed on the 36th and you recover your 35 failures to break even. The chance is 61.71 percent you'll score within 36 spins.

A different approach to consider with a \$100 bankroll is to play through 20 spins even though you may win one or more along the way. The chance you'll lose them all and finish \$100 in the hole doesn't change; it's 58.66 percent. Winning exactly one, a 31.71 percent probability, nets you \$175 minus \$95 or \$80. The chances of winning two or three and finishing ahead by \$260 and \$440 are 8.14 and 1.32 percent, respectively. More hits are possible, but the probabilities get remote. Government bailouts have made trillions into familiar, if not comprehensible, figures. So you might appreciate knowing that the chance of 11 hits in 20 tries, which will put you \$1,880 ahead, is one out of 1.8 trillion.

Maybe you don't think \$80, \$260, \$440, or even \$1,880 is enough to justify risking \$100. Making the next mortgage payment won't delay foreclosure very long. You want to pay off the balance; a giant jackpot on a major moolah slot machine or table game bonus be would do it. Some assumptions are required for these cases because you get intermediate returns while waiting for the biggie so a \$100 stake might get you 300, 400, or more tries at \$1 each before you fulfill your fantasies or face your fate.

Picture a jackpot game in which you have a single-round chance equal to one out of five million to win \$100,000 with a \$1 wager. Make believe you have \$100 to risk and, with the low-level returns, you have 400 tries. The chance of striking it rich with this much play is one out of 12,566. Were the low end hit rate to be higher so \$100 gives you 500 rounds, the chance of a jackpot would be one out of 10,083. How much brighter would your prospects be if you scraped up \$200 and got 1,000 spins? The chance of a jackpot would increase to one out of 5,162. Almost – not quite – twice as good, but hardly money in the bank.

A key point is that although your outlook improves with more rounds, it's never certain. Return to double-zero roulette. The wheel has 38 grooves, each with a probability of 2.63 percent. Betting \$1 on every spot in the same round, the chance of a hit is 100 percent (of course, it'll lose \$2 net). But, with \$1 on a single spot for 38 spins, the chance of winning exactly once is 37.28 percent, at least once is 67.70 percent, and none at all is 36.30 percent. The simultaneous and sequential situations differ, although many folks miss the distinction. It's as the coupleteer of clarification, Sumner A Ingmark, cleverly commented:

Ask precisely what you want to know,
Else the answer mayn't be apropos.

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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.