The Gambling Example
Considering gambling illuminates the mathematical representation of expected benefit. Suppose that we meet every day, and you pay me $2 to flip coin. If the coin comes up heads, I pay you $3; if it comes up tails, I pay you nothing. If we do this long enough, I will come out way ahead. The reason is that the probability of heads and tails is the same: 0.5. That is, over the long run, there will be as many heads as tails. Each time we flip, you pay me $2, and half of the time, I pay you $3. So in 100 flips, I get $200, and you get $150.
This is a bad bet for you. The expected benefit calculation captures the badness of the bet. The value to you of the coin coming up heads is $3. The probability of this is 0.5. So the expected benefit is $1.50. But you are paying $2 for this benefit. It is irrational to spend $2 to get $1.50.