How to Bet on Anything (If you want to win)

Introduction

You’re in Las Vegas for a business convention when you’re approached by a funny-looking guy on the street.

It’s me, and I’m holding an ordinary, 6-sided die. I propose a game, as follows:

You roll the die, and either:

1. Receive the amount shown on the die in dollars, or
2. Roll the die again, and forfeit this roll

You can roll the die at most 3 times, and you only get to keep your last roll. If you re-roll, you forfeit the previous roll.

“I’m not an artist, so bear with me” — Every math teacher, and now me

Here are 2 examples:

1. You roll a 4, re-roll a 5, and take the $5.
2. You roll a 3, re-roll another 3, then re-roll a 2. You must take the $2 because you used up all 3 rolls.

Now that you understand the rules, here’s the catch: it costs $4.50 to play my game. Do you play? If so, what’s your strategy?

Take a moment to think about your strategy before reading the next section. Better yet, take a moment to ponder whether this game is actually a useful abstraction for countless other real-world scenarios. Oops, I forgot to say *spoiler alert*.

Strategy