This week's Riddler Express draws from the game show Who Wants to Be a Millionaire:
You are a contestant on “Who Wants to Be a Riddler Millionaire.” You have already made it to a late round: You could walk away right now with $250,000. But there are two potential questions still to go that you can try to answer. You could earn $500,000 if you get one right and then walk away, or $1 million if you nail them both. If you attempt any answer and miss, you go home with $10,000.
The 50/50: The host reduces the four possible answers to two; one of them is the correct one and the other is randomly chosen from among the other three answers.
Ask the Audience: The studio audience submits their own guesses.
You know historically that the correct answer will be chosen by the plurality 50 percent of the time; while 30 percent of the time the right answer finishes second; 15 percent third; and 5 percent last. Additionally, if there are only two answers available to the audience, they pick the correct one more often 65 percent of the time.
Aside from leaving with the $250,000 you already have, there are four options that I can think of here:
- Option 1: use both hints on the first question, then quit
- Option 2: use both hints on the first question, guess on the second
- Option 3: guess on the first question, and use both hints on the second
- Option 4: use one hint on each question
I simulated the outcome to approximate the values for each of these scenarios, and Option 1 ($328,219) is the only one that provides a meaningful improvement in your expected value. Using one hint on each question (Option 4) increased the expected value slightly ($257,631) while Options 2 ($193,251) and 3 ($170600) both have an expected value that is less than just walking away. Of course, as a very smart friend pointed out, expected value works for the game show, who - if they last long enough - has many contestants (similar to a casino). As a player you only get one chance at this, so you have to consider whether the increase in expected value of ~ $80,000 is worth the risk of losing almost a quarter million dollars. Results are in the tables and chart below, and the Jupyter Python code is on Github.
| Option | Expected value |
|---|---|
| Option 1 | 328,219 |
| Option 2 | 193,251 |
| Option 3 | 170,600 |
| Option 4 | 257,631 |
Nice! That was a fun discussion.