Given that sports betting has come to North Carolina I am going to revisit the topic and derive some general formulas. The take home message is that while you might think you could make money betting against people who know less than you do. It is just like going to a casino.  The typical moneyline wager looks like

Fav(orit)e  -B      (Under)Dog +A

What this means is that you have to bet $B on Fave to win $100, while if you bet $100 on Dog you win $A. Let p be the probability Dog wins.

For the bet on Fave to be fair we need 100(1-p) – Bp = 0 or p = 100/(100+B)

For the bet on Dog to be fair we need -100(1-p) + Ap= 0 or p = 100/(100+A)

In practice A< B so if 100/(100+B) < p < 100/(100+A) both bets are unfavorable.

For a numerical example, in the sweet 16 of the 2024 NCAA tournament we saw the bet

Marquette -300, NC State + 230, so both are unfavorable if

0.25 = 100/400 < p < 100/330 = 0.303

In this particular case NC State won, which is not incredibly unexpected since when you roll two dice then the probability of a total of 9 or more has probability 10/36 = 0.2777

Another way of looking at this is through the money bet. If a fraction x of people bet on Fave then

When Fave wins the average winnings are 100x – 100(1-x) which is < 0 if x < 1/2

When Dog wins the average winnings are -B x + A(1-x) which is < 0 if x > A/(A+B)

In our concrete example both bets are bad for players if 0.434 = 230/530 < x < 1/2

While the sports book has no control over the probability that Dog wins, they can control the fraction of money bet on the favorite by adjusting the odds over time, or using their knowledge of previous bets to choose good values of A and B.

When A/(A+B) < x < ½ then the sports book has an arbitrage opportunity. They will make money under either outcome. In the theory of option pricing it is assumed that arbitrage opportunities do not exist (or if they do are short lived), and based on this one can derive what is the right price for a “derivative security” such as a call or put option based on the stock prices.

If you want to learn about this look at Chapter 6 of my book Essentials of Stochastic Processes which you can download in pdf form from my web page. https://services.math.duke.edu/~rtd/EOSP/eosp.html

My goal here is to make two points (i) the sports books are not gambling: they have things set up so that they win money no matter what the outcome is, (ii) even though you are gambling against a group of people and it seems that you can win money if you are smarter than they are, that is an illusion. Like casino gambling, things are set up so that all of the bets are unfavorable. So like the TV commercials say look at sports betting as a way of enhancing the fan experience not as a way to make money. However with betting available 24-7, in game parlays and bonus bets designed to get you used to betting a lot of money, this like the lottery, is a scheme designed to take money from people who are gambling with money they can’t afford to lose.