Right now, I have 1€. And I know exactly how to invest that euro to make profit, but I have 3 options:
- Invest the euro, and win exactly 1€ (I'd have 2€ then)
- Invest the euro, and earn some money between 0,5€ or 1,5€ (I'd have an amount between 1,5€ or 2,5€)
- Invest the euro, and win either nothing or 2€ (I'd have an amount between 0€ or 4€)
After investing that euro one time, I would be investing one euro again and again.
So, what's the best option of these 3 to earn some money?
Are they proportionally equal, and happens the same with all the different values?
If that's true the winnings would be just the average of both min and max values.
(Not real try to earn money)
If losing the euro doesn't stop you from participating in the future, after a long series of bets you will (on average) be the same place in all three options. The expected value of each is +1 euro. After a large number of plays you will converge on a normal distribution with mean the number of plays and some variance. Option 1 gives you no variance, option 2 some and option 3 even more. Many people prefer the sure thing, except they irrationally favor low chance high payout events (see the lottery).