Expected monetary value

- Suppose you're considering entering a team into a 16 team basketball tournament. The competition is a bit expensive though. The entry fee is $750, but most of the money, $10,500 will be awarded back to the teams as prizes for winning. There are four rounds. Every time your team wins a game, your team wins $700. The big question, of course, is should your team enter the tournament? For this, we can use a variation of the mean for a discrete probability distribution.

It's pretty much the same thing except since we're using money, it's often called the expected monetary value. So let's figure out the expected monetary value for a team that enters this tournament. This is a competition, so the expected monetary value is really dependent on how good your team is. But let's begin with the premise that all teams are equal. Effectively, each game is a coin toss. So first, let's calculated the expected monetary value for each team assuming they are all equal.

Let's use this table to see how the tournament would play out. Only one team can win the championship. That team will win four games. One team will lose the championship game, but they will win three games to get there. Two teams will only win two games. Four teams will win only one game, but eight teams will lose in the first round, and thus, they won't win any games. With 16 teams in the tournament, we can now calculate the relative frequency for each of the win totals, and if a team wins $700 every time they win a game, here are the total winnings awarded for four to zero wins.

We now multiply the winnings times the relative frequency for each win total, so for four wins, we multiply the $2,800 in winnings times the relative frequency of 0.0625, or 6.25%. We do this for every win total. We add up these products, and we get $656.25. So that's our expected monetary value, right? Not quite.

Don't forget, every team had to pay $750 as an entry fee, so $750 minus our $656.25 in expected winnings tells us the expected monetary value for the average team is a loss of $93.75. But let's say your team is better than the average team. Let's say a professional basketball consultant has looked at the likely field of teams and has estimated that your team as a 5% chance of winning all four games, a 15% chance of winning three games, a 25% chance of winning two games, and also a 25% chance of winning one game, and finally, a 30% chance of losing the very first game.

What would be the expected monetary value for your team? Again, for each discrete outcome, we multiply our winnings times the probability. We add up all of our products, and now we see that even with our $750 entry fee, our team should get $980 in winnings for a total expected monetary value of $230. So, as you're trying to figure out what your commissions for next year might be, what the cost of a big corporate project might be, or perhaps as you consider the sales probabilities for next quarter, you can use expected monetary value to help you understand the possibilities, consider your options, or you might even use expected monetary value to persuade others about a money related decision.