The correct answer is that there is a better strategy than always refusing the bet. Namely: choose either Red or Blue beforehand and bet Tails only when you see that the room is in this color. This way the Beauty bets 50% of time when the coin is Heads and every time when it’s Tails, which allows her to systematically win money at 2:3 odds.
You place $200 down, and receive $300 if the coin was indeed tails.
If the coin toss ends up heads, you have a 50% chance of losing $200 - expected utility is $-100.
If the coin toss is tails, you have a %100 chance of gaining $100 - expected utility is $100.
So you end up with expected 0 utility.
The point stands, but the odds have to be better than 2:3.
You place $200 down, and receive $300 if the coin was indeed tails.
If the coin toss ends up heads, you have a 50% chance of losing $200 - expected utility is $-100.
If the coin toss is tails, you have a %100 chance of gaining $100 - expected utility is $100.
So you end up with expected 0 utility.
The point stands, but the odds have to be better than 2:3.