Would you pay $10 for a 1% chance to win $1000? Different people might evaluate this gamble differently depending on their risk tolerance and life experiences. What about $10 for a .1% chance to win $10,000? From a mathematical standpoint, the odds are the same, but emotionally we react differently. Expected value is one framework to think about these choices.
At the simple level, as with the two scenarios, expected value is simply a multiplication of probability or chance vs the dollar amount that can be gained or won. The results are both $10, so therefore a $10 bet is worth $10 of expected value. There’s nothing to gain, so it’s not worth it.
Obviously, if the calculation was less than $10, you shouldn’t do it. This is the case with pretty much all lotteries and casino games – that’s why the house always wins, yet people continue to play. This is partly because they don’t understand the expected value of .000001% x 1,000,000. We’re wired to focus on the million dollars without calculating that the expected value is actually only $1.
The wise approach is to take all bets where the expected value is greater than the amount bet, so if the expected value of a $10 bet is $11, you should always take the bet (unless there are other considerations such as time value), yet oddly enough, some people might not perceive the small gain as worth the risk.
Our emotions and perceptions of probabilities and big numbers can be distorted depending on how these choices are presented to us, but it is possible to take all these situations and apply a common approach, expected value, to equalize them so that you can make better money decisions.