Door No. 1, 2 or 3? Does it matter?

I ran across a brain teaser the other day that I found to be a stark reminder of how people often approach and form opinions about unfamiliar issues they do not recognize as unfamiliar. Er…what?. Let me continue.

It’s called the “Monty Hall problem.” Monty Hall was the creator and host of the original “Let’s Make a Deal,” the longrunning daytime game show that featured people dressing up in crazy costumes while jumping around and scream hysterically.

The “Monty Hall problem” would present a contestant (who was dressed up as some kind of character — let’s call them “Barney”) with three doors. Behind one would be a cool prize, often a car, and behind the others would be booby prizes, called “zonks.” Barney would be offered a choice between the three doors and would choose one -- for our purposes, door No. 1. Before revealing what was behind door No. 1, Hall would reveal the contents behind one of the remaining doors, say door No. 3. Of course, it would not be the car, as Monty knew which door had the car and would never tip his hand. Now Monty would offer Barney the option to change to door No. 2.

What to do? Do you stay with door No. 1, or do you switch to door

No. 2? Or does it matter? It’s a conundrum, and one that spans all levels of education and affluence. About 80% get it wrong; interestingly about the same number who guess wrong about when to take Social Security. Which side do you come down on? Do you think it doesn’t matter, or do you believe it does? Do you believe Barney should keep door No. 1 or switch to door No. 2? It really doesn’t matter, right? It’s a binary choice. There are two doors. Behind one is a car and the other a zonk. So, it’s 50/50. Might as well stay with what you have. Or should you?

Assuming we cannot come to a reasoned answer (we can, and will, in a moment), is there a strategy that will allow us to default to the correct answer even if we cannot recognize it? Yes, there is. In this case, it means you should change, which is also the correct answer when you fully understand the problem.

Let’s first deal with why this strategy can get you to the right place even if you guess wrong, and then we will solve the puzzle. If it truly is a 50/50 proposition, it doesn’t matter if you change or not, right?

Two choices with two possible outcomes. Fifty/ fifty. Even Steven. A wash. So that leads directly to the one right answer … you should always change. Why? Because there is always that extremely small, minute chance that it does matter. That changing might be the correct choice. So you have two potential outcomes: It doesn’t matter, or it does. By changing, you cover both possibilities.

But wait. What if staying was the statistically correct choice? It can’t be. It’s always going to be either it truly doesn’t matter, or that changing can give you an unknown and unlikely benefit.

And this leads us to the actual correct answer: You should always change.First, let’s look at what this choice is not. It is not a binary choice. It never was. It began and remains a choice between three options. You only get one choice: door 1, 2 or 3. That creates two “sets,” if you remember your sixth-grade math: “unchosen” and “chosen.” Unchosen, which has two doors, has twice the probability of being correct, right? Chosen has a 1 in 3 chance of being correct, and unchosen has a 2 in 3 chance, which means unchosen is twice as likely to be hiding the car. However, unchosen will also include a zonk 100% of the time.

Now, Monty opens door No. 3, revealing the zonk, knowing it will confuse you into thinking the odds have gone from 1 in 3 to 1 in 2. But has it? We already know at least one of the unchosen doors has a zonk behind it, and all Monty has done is show you which one it was, thereby eliminating it as a choice. It’s still a member of the unchosen set, which has a probability of 2 in 3 of having the car, just with one door open. It did not in any way change the original odds: unchosen still has a 2:1 probability over chosen.

You are still being offered two doors over one door, which always had a better chance of having the car. You simply know which of the two doors is not the winner, and that has been removed as a choice, but does not change the odds that door No. 2 still has a 2 in 3 chance of having the car behind it.

The correct answer is always going to be to make the switch. Either it doesn’t matter, and switching can’t hurt you, or it does matter, and switching helps you. Either way, switching is the best choice, regardless of how your mind perceives it.

So what choices are you making in your retirement plans that have similar gotchas? Well, Social Security elections, pension elections, how to allocate your investments, what accounts to spend first and how to invest your money leading up to retirement all have these types of conundrums.

Fortunately, there are ways both to derive the correct answer, or back into it like in the above case. In either case you end up making the correct decision. And often, just like with the Monty Hall problem, making the correct choice can significantly improve your odds of success.