This Maths Riddle Has PhDs Scratching Their Heads

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats.

You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat.

He then says to you, “Do you want to pick door No. 2?”

Is it to your advantage to switch your choice?

Parade Magazine

You may recognise this brain teaser as the famous Monty Hall problem. Named after the former host of Let’s Make a Deal, this baffling puzzle has confounded people for nearly 45 years.

There is a correct answer…and it’s likely you got it wrong.

And if you got this wrong, it’s possible you’re thinking about certain financial decisions the same way.

We’ll reveal those financial implications in a moment…but first, allow me to explain this riddle.

I first heard of the Monty Hall problem in uni from my engineering major roommate. As you’d hope from an engineer, he was a smart guy (He later got headhunted by the oil industry and designs the complicated structures that make up an oil pipeline).

He posed the problem to me the same way…you start off with three doors.

Source: Taylor Kee

Behind two of the doors are goats, but behind one of the doors is a brand-new Toyota Hilux. At this point, you have no information…so you’re essentially gambling with a one-in-three chance of winning the truck.

One-in-three chance…that’s important.

So you pick door No. 1 at random.

Then the host, who does know which door is hiding the car, opens another door revealing a goat. You think he’s just spicing things up by offering you a clue, but in reality he’s upped your odds significantly.

Source: Taylor Kee

So you still don’t know what’s behind doors 1 and 2, but you know that door 3 houses a goat.

The host offers you the chance to change your choice of door. You had picked door No. 1 originally, but you could change to door No. 2 if you want.

Instinctually, your reaction might be like mine — ‘I still don’t know which door hides the car…so it doesn’t really matter. It’s 50/50, right? I’ll stick with my original guess of door No. 1.’

But like me, you’d be completely wrong.

The solution, in fact, is to always switch your door when the host offers you another chance because by switching, you give yourself a 2/3 chance of winning the Hilux, whereas staying only gives you a 1/3 chance.

That’s right…statistically the odds are in your favour to switch to door No. 2 after the host reveals the goat behind door No. 3.

Here’s how my roommate explained it to me: instead of three doors, imagine there were a million. Behind only one door is the car. The rest hide goats. Your first guess is a one-in-a-million shot. Pretty low chance of guessing it at that point.

Then the host says, ‘Hold on’ and opens 999,998 doors to reveal goats…leaving only your original choice and one other door unopened.

He then gives you the chance to switch doors. Would you do it?

Of course!

Originally, you probably picked a goat because, remember, your odds were one-in-a-million for your first choice. And now the host, who knows where the car is, has eliminated 999,998 of the other options.

The odds of the other door hiding the car is 999,999/1,000,000 or 99.9999%.

By volunteering information, the host has filled in a lot of blanks in the equation…essentially giving you the answer.

Now, if you’re still confused or think that the odds are 50:50 for the second choice, don’t feel bad. Only 13% of folks who try this puzzle make the choice to switch doors…and probably only a small percentage of those understand why it’s better to switch.

The person who first published the Monty Hall Problem, Marilyn vos Savant, was met with uproarious criticism, even from the statistics field. Here’s what Scott Smith, PhD at the University of Florida said to vos Savant:

You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!

And that came from someone with a PhD in maths!

There are two lessons learned from the Monty Hall problem:

  1. Always switch doors if you end up on Let’s Make a Deal.
  2. With more information, you improve your chances of success.

That second golden nugget really hits home for investors. We’re constantly trying to improve the odds of our investment choices. We’re trying to pick the Hilux out from a million goats.

What we try to do here at Money Morning NZ is open as many doors as we can…to provide you, the investor, with improved odds of finding lucrative investments.

Never stop seeking more information. It pays off.


Taylor Kee
Editor, Money Morning New Zealand

PS: For those still scratching their heads on this one…here’s one more way of looking at it. Assume you’re dealing with the original three-door problem, with the car behind door No. 1. You make a first decision, the host eliminates a goat door, then you get the opportunity to switch. Here’s how it could play out:

  1. Option 1: You choose door No. 1 – you lose by switching.
  2. Option 2: You choose door No. 2 – you win by switching.
  3. Option 3: You choose door No. 3 – you win by switching.

So you have a 2/3 chance of winning the car by switching doors!

If you’re still adamant that it’s 50:50, change my view by reaching me at

Taylor Kee is the lead Editor at Money Morning NZ. With a background in the financial publishing industry, Taylor knows how simple, yet difficult investing can be. He has worked with a range of assets classes, and with some of the world’s most thought-provoking financial writers, including Bill Bonner, Dan Denning, Doug Casey, and more. But he’s found his niche in macroeconomics and the excitement of technology investments. And Taylor is looking forward to the opportunity to share his thoughts on where New Zealand’s economy is going next and the opportunities it presents. Taylor shares these ideas with Money Morning NZ readers each day.

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