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# Probability, Frequency, and Luck

15 October 2012

Luck, good and bad, in casino gambling arises fundamentally from differences between the probabilities or expected rates of bets winning, losing, or pushing and the frequencies at which these outcomes actually occur. This, because payoffs for wins are set in advance based on the probabilities while bankroll changes are determined during the course of the action by the frequencies. Good luck is characterized by frequencies higher than the theoretical probabilities; conversely for bad luck. How good or how bad depends on the magnitude of the difference.

Picture the archetype of all gambling, the flip of an unbiased coin for even money. Ignoring coups that result in coins disappearing down sewer grates or landing on their rims, bets on one face or the other would have 50 percent – one out of two – prospects of winning.

Pretend you flip twice, betting on heads both times. A literal interpretation of the 50 percent probability would get you 0.5 x 2 or one instance of heads in two tries and therefore breaking even. This, in fact, could be your frequency and monetary outcomes.

But the frequency actually obtained might also differ from the probability figure. You could hit zero percent – zero out of two, in which case you’d be two units behind and cursing your bad luck. Alternately, you could hit 100 percent – two out of two, finishing two units up and chortling over your good luck – if not your remarkable gambling skill. Were you to flip four times, your expectation based on probability would be to obtain 0.5 x 4 or two instances of heads in four tries, breaking even. The actual frequency could be from zero through four heads. The rate might accordingly be zero percent – zero out of four – 50 percent less than the theoretical value, 25 percent – one out of four – 25 percent less than the theoretical value, 50 percent – two out of four – equal to the theoretical value, 75 percent – three out of four – 25 percent greater than the theoretical value, or 100 percent – four out of four – 50 percent greater than the theoretical value. Monetarily, these outcomes would yield very bad luck with a four-unit loss, moderately bad luck with a two-unit loss, neutral luck with a break-even series, moderately good luck with a two-unit win, and very good luck with a four-unit win, respectively.

Real casino games differ from the indicated coin flip, in essence, only because the include a mechanism for the establishment to operate as a business by earning a fee on the action. The mechanism, the house advantage or edge, is implemented by means of an offset between the odds that a bet will win and the payoff when it does.

As an illustration, a double-zero roulette wheel has 38 grooves 18 red, 18 black, and two green. In most casinos, a bet on red pays even money but the probabilities are only an18 out of 38 – just under 47.4 percent – of winning and the complementary 20 out of 38 – 52.6 percent – of losing.

Make believe you bet on one spin and the ball lands in a black groove so you lose; you have bad luck because the frequency of wins you experience of wins is 0.0 percent – 47.4 percent less than the theoretical probability. Instead, assume the ball lands in a red groove so you win; you have good luck because the frequency of wins you experience is 100 percent – 52.6 percent more than the theoretical probability. Note that, because of the edge, you had to have more good luck to win than bad luck to lose – a difference of 52.6 as opposed to 47.4 percent.

You can’t predict the frequency of wins in a session. But the chances associated with various frequencies can be determined, given the theoretical probabilities. For the four-flip coin session, where the probability of winning on any coup is 50 percent, chances of the frequency being from zero to four out of four are as given in the nearby table.

Chances of experiencing a various frequencies of heads in four flips of an unbiased coin

Frequency             Net       Chance
zero out of four     \$4 loss     6.25%
one out of four      \$2 loss    25.00%
two out of four   break even    37.50%
three out of four     \$2 win    25.00%
four out of four      \$4 win     6.25%

The table shows that as the amount of luck encountered – good or bad – increases, the chance of its having done so decreases. Which helps to explain how good players decide when to quit with a profit rather than press on, or cut their losses rather than expose themselves to a downside risk they may later regret taking. Here’s how the poet, Sumner A Ingmark, described these opposing situations:

To know you’re where you sought to be,
Or somewhere you want not to be,
Means knowing where you ought to be.