Basic concept of utility: utility of expected value vs expected utility

I'm new to the concept of utility and I'm struggling to understand an important idea. Say we have some bet. I don't understand how the utility of the expected value of the bet differs from the expected value of the utility of the bet. In my mind, both correspond to what I would expect to get out of the bet on average, scaled to take in account my preferences for different monetary values.

Perhaps a simple example showing how exactly the two ideas are different would help. Thanks!

This situation can happen when an individual is risk adverse. Take for example a fair coin flip bet--if heads, the person wins 1 dollar, if tails, the person loses a dollar. Let's say for this person, gaining the dollar has a value of 1 utility unit, neither gaining nor losing has a value of 0 utility, and losing a dollar has the utility of -2 utility units. In this case, the expected value from the flip is 0, and the utility of the expected value is 0 utility. However, the expected value of the utility is $0.5(1)+.5(-2)=-0.5$, since the utility function has the person being risk adverse. So in this case the expected value of the utility does not equal the utility of the expected value.

Imagine that you are on vacation away from home and need to get back home in a hurry because a loved one absolutely needs you. Imagine that you have \$1000, of which \$950 is needed to purchase your plane ticket home. A stranger offers you a bet : heads you win \$1000, tails you loose \$100.

Do you take the bet?

On pure expected value, you are looking at an expected gain of $(1000-100)/2=450$. Say, the utility of this is 450 utility units (u's). Note that your expected wealth after the bet is \$1450, more than enough to get you home.

In terms of utility, the gamble is between winning \$1000, say 1000 u's, and not making it home, say -1000000 u's. The expected utility is $(1000-1000000)/2=-499500$u's.

The correct way to approach this is expected utility, not expected gain. You do not take the bet.