Anthony Dee

While the study of mathematics has a wide variety of uses today, one of the earliest motives of mathematical study was to make a profit from gambling. Research in gambling led to the development of probability theory, one of the more popular disciplines of mathematics today. Probability theory is, as the name states, the mathematical study of probabilities, used in applying math to random variables and random processes in an attempt to make determinations from them. While perfect predictions are impossible, probability theory is utilized by casinos to ensure large profits, and can be applied to analyzing the lottery as well.

You often hear stories about someone “winning big” at a casino or in a lottery, but this is an expected result of the amount of people that spend money gambling or buying lottery tickets. In fact, it’s more beneficial for casinos to have the occasional big winner, because it inspires people to keep going. Unlike alcohol, tobacco, or drugs, addiction to gambling does not require any chemical intake; simply the possibility of having a huge payout is enough to squander large sums of money. This indicates that addiction to gambling is innate in people– but utilizing probability well, unfortunately, is not.

Casinos make use of people’s lack of ability in calculating probabilities to their advantage. Huge payouts (in the thousands or more) occur as a result of larger bets, but the chances of winning are extremely slim. While they lose a decent sum of money occasionally, this is easily paid off by all the other gamblers that have lost money to the casino– contributing towards profit. Take the simple example of a dollar slot machine; say the payout is $500, and there is a \(\frac{1}{2000}\) chance of winning. When someone wins $500, people are naturally attracted to the machine for a chance to win big as well. However, the probability states that for one person winning $500, on average the other 1999 of the 2000 attempts will lose the $1, so the casino makes a $1499 profit after they pay the winner– three times more than what was lost.

In addition, casinos often pay less winning wages than what is deserved by “true odds,” which pretty much guarantees a profit. Essentially, when someone places a bet on a certain result, the bet is multiplied by some fixed number as the payout. A simple example of this is betting on the outcome of a dice roll; for unrigged dice the chances of winning is simply ⅙, meaning the losing outcome happens ⅚ of the time. In other words, on average you will lose ⅚ of what you bet, five times more than the ⅙ you keep, and that is gained by the casino. In the event of a win, it follows that paying five times the bet would let both sides break even. However, casinos guarantee a profit by paying with a multiplier less than the “true odds.” Take for example, people bet $10 six times on the outcome of a dice roll. A perfect probability outcome would be one win and five losses, meaning the casino would gain $50 and pay out $10 times a multiplier. If the multiplier was five or greater, then on average they would make no profit, but this is often accounted for, and the multiplier would be reduced to three or four in order to guarantee a profit. Similar logic is used for lotteries.

Optimal stopping theory is a concept in mathematics that concerns choosing a time to take an action, in order to maximise the reward and/or minimize the cost. This can be applied to statistics, economics, finance, and other fields. Optimal stopping theory plays a part in choosing parking spaces, selling houses, option trading, search theory, and of course, gambling. However, one of the best examples of this concept is the secretary problem.

Imagine an administrator with the goal of hiring the best possible secretary from some number of applicants. Applicants are interviewed individually in random order, and a decision has to be made directly after the interview; once an applicant is rejected they cannot be recalled. The administrator ranks the applicants that have been interviewed, but does not know anything about those who weren’t interviewed. It turns out that maximizing the chances to pick the best secretary is related to Euler’s number, \(e\), resulting in the \(\frac{1}{e}\) law of optimal strategy.

Say you have 15 potential candidates for the secretary position. Since you are ranking the applicants as they go, you will eventually form a benchmark for later candidates. If you have interviewed two people, that means you can still choose from 13 applicants, but you lack sufficient information to create a good benchmark. On the other hand, if you have a sample size of 13 interviewees, you have sufficient information to judge the other candidates by– only you now have a limited amount of 2 to choose from. So it seems obvious that you want to choose the perfect sample size, not too big and not too small. But what is it?

Amazingly, the best way to get a perfect sample size is to reject the first \(\frac{n}{e}\) (rounded up) interviewees (*n* being the total number of applicants) and hire the next. In the example with 15 applicants, the administrator should reject the first six interviewees and hire the seventh. This results in a roughly 37% chance of success, which believe it or not, is the best you can get. Thus, even with optimal stopping theory, you are unlikely to succeed.

While optimal stopping minimizes your loss, it is clearly not perfect, and gambling is by no means the most reliable or risk-free way to earn money. Math is not on your side.

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