> Matthew Frazer Case #67 - Chance Science & Technology

The Deal of Your Life

September 29, 2023

Image: Mostafameraji, Wikimedia Commons

Matt Frazer

Monty Hall was the host of the American TV game show ‘Let’s Make a Deal’ which first aired in 1963. Contestants on the show played games of chance to win cash and other prizes. The game show inspired a letter to the American Statistician magazine in 1975 which posed the now famous ‘Monty Hall Problem’.

In the standard formulation, a contestant is shown three closed doors. Behind one is a major prize, like a car. Behind the other two are consolation prizes—typically goats! Without knowing what is behind each one, the contestant must pick a door. After they have selected, to build tension, the host will open one of the doors they did not pick to reveal a goat. The contestant will then be given a choice to either keep their original selection and receive whatever prize it hides, or switch their choice to the now only one remaining door. What compelling viewing!

Depending on your relationship with statistics, gambling, or afternoon television, you may have strong feelings about what the hypothetical contestant should do in this scenario. What is their chance of winning by keeping their original choice? Does it improve by switching?

That fact that you are reading about this some half a century later might tip you off that mischief is afoot. Best to stop shouting at the TV or smugly deriding the intelligence level of the average TV game show contestant. Thousands of readers of the American Statistician wrote in (proudly signing with ‘PhD’) stating the obvious answer that switching does not improve the contestant’s chances at all.

Except this obvious answer is demonstrably wrong.

Experience with a multitude of high school students tells me it is unlikely I will entirely convince you of this in the few words I have here, but a concise explanation goes like this: The original door indeed has about a 33% chance of concealing the prize. Which means that the other two doors have, between them, about a 67% chance of doing so. When the host opens one of the two unchosen doors, he reveals some information about the location of the prize. If one of the unchosen doors did contain the prize, he would not open it, as that would spoil the tension. So he always opens a goat-door, meaning the 67% probability is now ‘concentrated’ onto the one remaining door. By switching doors, the contestant doubles their chance of winning.

Aside from being a great ‘gotcha’ conversation starter for the nerdiest of parties , why am I telling you this?

Probability is a function of not only the possible outcomes in a given situation, but also the information available to us about the situation.

Will it rain tomorrow? If you live in a windowless box and stay off the Weather Channel, perhaps you’d have to assign that a blind 50% chance. Look out the window to see the sun shine and maybe you could refine that to 10%. Gather detailed meteorological data for the entire continent and you might have more confidence in a number like 73%. With enough information, your answer could approach either 0% or 100%—near certainty. But for all your learnings, nothing in the sky has changed. Only your perspective, as you convince yourself that the matter of tomorrow’s precipitation has already been predestined.

Much ink has been spilled elsewhere on the matter of whether God plays dice with the universe (see Paul Ewart’s article in this edition, for example). But if God’s omniscience grants him the perfect perspective on every outcome, it could be simultaneously true that he has predestined every action in all creation, and that we mere contestants must deal with uncertainty.

Of course, this doesn’t mean we are at the mercy of that uncertainty in all our life decisions. God has generously given us more information about how to make the biggest decision we face than the Make A Deal host reveals to his game contestants. The host leaves even the shrewdest contestant with a 33% chance of missing out. But through Jesus, God—who not only knows the right answer but wants us to choose it—tells us outright which door leads to the greatest prize.

I tell you the truth, whoever hears my word and believes him who sent me has eternal life and will not be condemned; he has crossed over from death to life. (John 5:24)

It’s true that we have to take God’s word for it until the door is finally opened and we experience first-hand what we now see through eyes of faith. But if we do, Jesus assures us we will be separated from the goats once and for all.

Matt Frazer graduated from UNSW in Engineering and Science. After a time programming machines he switched to the more challenging task of programming teenagers, and now teaches high school Mathematics and Computing.