Will You Make Money?

No one can win at roulette unless he steals money from the table while the croupier isn’t looking.

— Albert Einstein (possibly)

The development of probability theory is historically linked to attempts to understand games of chance, especially ones in which money was involved (see for example, here). Sometimes betting money on an uncertain outcome falls under the name of gambling; other times it’s dignified with the name investment or “smart business decision.” But regardless of the label, there are smarter and less smart ways to play money games.

Battle of the Bills

Let’s recall a distinction we made earlier in this course about deterministic and stochastic, or random, processes. This time, we’ll think about two different bets you make with your friend. In the first bet, you and your friend are debating whether it was Bill Paxton or Bill Pullman in the movie Apollo 13. To make the game interesting, you bet two dollars. You look it up on the internet, and find that it was indeed Paxton. One of you wins. Do you feel the need to check again? Probably not. This particular question, although you may not have known the answer for sure, has only one possible answer, no matter how many times you check.

Now consider another bet, this time for three bucks! You and your friend are walking down the street debating the “merits” of mint chocolate chip vs. cookies and cream as ice cream flavors. You claim that mint chocolate chip is the more popular flavor, and decide to ask the first passer-by which flavor they think is better. Suppose they don’t just ignore you, thinking you’re a nutcase, and they answer cookies and cream. Are you satisfied with this one answer? Or do you feel the need to ask another pedestrian? And how many?

We might say that the variable “BPA”, which stands for “which Bill P. starred in Apollo 13?” has a deterministic answer. It is always the same. But the variable “MCCoCAC” which stands for “mint chocolate chip > cookies and cream?” can take on different answer (one of two; no ties allowed) depending on whom we ask. Because it is a random or stochastic variable, we have to talk about it using different terms. We might say something like, what proportion of people (in this neighborhood, say) prefer cookies and cream? Or what are the chances that the first person we ask will express that particular preference.

This may all sound like silly bets that are really just games between friends. But people make small and large money bets all the time, in everything from business and life decisions, to recreational games. In this module, we explore probability calculations that inform things like advertising, airplane booking, the job market, and march madness.