The Three Elements of a Bet

The unifying thread in this mixture of dice, cards, wheels, horses and athletes is the follow-ing. A trade or exchange is going on. The bettor is trading:

1. a known amount of money now in exchange for
2. a larger sum of money in the future,
3. contingent on the objective outcome of a pre-selected event.

In a way, a bet is like a contract, in which the bettor performs now by paying up, and the receiver of the bet (usually the house) promises to perform later, but only if a specific, ob-jectively verifiable event takes place. Note: Bets are never registered on a promise to pay, or credit. In every instance, the bettor must support his or her bet with cold, hard cash – or its equivalent -- now.

Whether or not a bettor is making “a good bet” depends upon the bettor’s personal goals. If one of the goals is to bet only on propositions with a positive expectation of winning, then a “good” bet will have a certain characteristic related to the relationship of these three elements.

One classic, but simple example, is the “prop” that I (the house) will pay you (the bettor) 5 to 1 on your flipping a fair coin three times and coming up with three heads. So if you give me $1 and flip a coin three times, I keep the dollar unless you flip three heads in a row. If that happens, I give you your dollar back and $5 more besides. Is this a “good bet”?

Note first of all, that had you not been told the amount of your risk (item 1) and the amount of your payoff (item 2) and the event that was to happen in the future (item 3), it would not be possible to assess the bet at all. It would be impossible to say whether the bet was good or bad.

In the example, the mathematical chance of flipping three heads in a row is 1 in 8. The house’s offer is a risk at 1 in 6. (Five to one is the same as one in six. The first version of how this is expressed puts the emphasis on the risk vs. reward. The second version includes the original bet in the payout).

To express exactly what is going on, the house odds are presented as if the physical expec-tation were 16.6% rather then 12.5%. This makes it easier to compare the two situations. Most people would consider the proposal ungenerous on the part of the house. Even if the bettor takes the bet and wins -- as he will do a certain number of times (about 12.5% of the time if the bet is repeated over and over) – in the long run, the bettor will lose. Why? On average the bettor will be paying out 16.7 cents for each flip, against an expectation of win-ning only 12.5 cents.

A “fair” bet might be 7 to 1 odds (or 1 in 8, which is the same thing). This mirrors the arithmetic that gave us an expected value of a 12.5% chance for three heads. In the long run, no one would be expected to win or lose anything. This is called “breaking even.”

A bet in which the bettor has a statistical expectation to win something would be 8 to 1 or better. At 8 to 1, the bettor is paying on average around 11.1 cents for an expected win of 12.5 cents, and that’s a positive expectation.

Well, that might have been difficult for the reader to get through. If you made it to this point, you may be asking yourself, “What does this mental exercise have to do with having fun in Las Vegas?” That is a good question. Here’s the answer.