If you’re risk-neutral, you still can’t just do whatever has the highest chance of being right; you must also consider the cost of being wrong. You will probably win a bet that says a fair six-sided die will come up on a number greater than 2. But you shouldn’t buy this bet for a dollar if the payoff is only $1.10, even though that purchase can be summarized as “you will probably gain ten cents”. That bet is better than a similarly-priced, similarly-paid bet on the opposite outcome; but it’s not good.
“You will probably gain ten cents” is not a summary of all the relevant information. If money and utility are interchangeable at this small scale (if you don’t desperately need an extra 10¢ but you’d prefer to increase the amount of money you have) then an expected utility calculation can tell you that it’s a bad bet — it comes out to an expected loss of 26.6¢.
“You will probably gain ten cents” is not a summary of all the relevant information. If money and utility are interchangeable at this small scale (if you don’t desperately need an extra 10¢ but you’d prefer to increase the amount of money you have) then an expected utility calculation can tell you that it’s a bad bet — it comes out to an expected loss of 26.6¢.