How To Be A Millionaire! (Three out of four times)

Thomas Wu


September 23rd, 2012

Applying Bayesian statistical techniques to Millionaire Hotseat.

Can statistical inference beat the system?                     

The room is dark. A spotlight shines down on you. This is it, the million-dollar question. You have no idea what the answer is so as always, when in doubt, you’ve guessed ‘C’.

You use your 50-50 lifeline.
It’s not very effective. C and some other letter remain.
Eddie McGuire flashes a cheeky grin in your direction, ‘you’ve picked C… Lock it in?’

After recently watching a friend of mine walk away with a tidy profit from his appearance on Millionaire Hot Seat, I decided to take a look at the classic lifelines that used to be available to contestants on the show.  Specifically, is the 50-50 lifeline really a 50-50 chance?

Bayes’ theorem is a relatively niche statistical technique used in economic modeling but is undoubtedly one of the most fascinating. It is a method that allows us to adjust our hypotheses in response to any new information that may come in. This is especially useful for real life situations that include constantly evolving data such as economics, biomedicine, or Who Wants To Be A Millionaire?

In this example, our potential ‘hypotheses’ of A, B, C or D have now been influenced through new data (the use of a lifeline) that has restricted our ‘hypotheses’ to either C and one other letter. It seems like a fair gamble right? Surely it’s a 50-50 shot?

Surprisingly, it turns out that changing your answer actually significantly increases your chances of guessing correctly.

While this may initially seem to be counterintuitive, consider the following tree-diagram:

Those who have encountered the infamous Monty Hall problem may be familiar with this outcome but the implications of this still blow my mind.

If we are left with the choice between our initial guess and a different one, then switching our answer will yield us a 75% chance of choosing correctly! Despite the fact that it appears to be a 50-50 situation.

Clearly, while Bayesian principles are not for the faint of heart or the dull-witted, the ability to incorporate new information into past results is undoubtedly vital for both practicing economists and aspiring Millionaire contestants.


Your final answer?
Change it up!

It’s guaranteed to work! (Three out of four times)

The views expressed within this article are those of the author and do not represent the views of the ESSA Committee or the Society's sponsors. Use of any content from this article should clearly attribute the work to the author and not to ESSA or its sponsors.

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