It’s a scenario every game show fan knows well. You’ve got three doors to choose from, and one of those doors will open to reveal a fabulous prize.
After you’ve made your choice (let’s say Door #2), our affable host Monty Hall plays Devil’s Advocate by opening one of the doors you didn’t choose (let’s say Door #1), revealing a goat or other lackluster result.
And then, Monty offers you a chance to change your mind. Will you stick with the door you initially chose, or will you switch to the other unopened door (Door #3)?
The average player sees two choices, Door #2 and Door #3, which on the surface sounds like a 50/50 shot, a coin flip. So would it surprise you to learn that people who switched from one door to the other doubled their chances to win the fabulous prize?
This is known as the Monty Hall Problem, an example of how statistics aren’t always what they seem, and it has puzzled people for decades.
It’s counterintuitive, isn’t it? I mean, you have two choices, so the odds should be 50/50. But you’re forgetting that third door that Monty eliminated. That third door makes all the difference, statistically speaking.
Let’s break it down. Your initial choice is between 3 doors, meaning you have a 1 in 3 chance of picking the correct door, and a 2 in 3 chance of picking the wrong one.
When Monty opens that other door, the odds haven’t changed. Only the number of options available has changed. Your door is still a 1 in 3 chance of being correct and a 2 in 3 chance of being wrong. But the remaining door now has a 2 in 3 chance of being correct!
So what appeared to be a coin flip between sticking with your choice and switching is now heavily weighted toward switching!
There have been several real-world tests of the Monty Hall Problem, and all of them have consistently shown that the people who switch were twice as likely to open the winning door!
The real puzzle here is how we fool ourselves. We take the numbers at face value — 3 doors become 2 doors, so a 1 in 3 chance becomes a 1 in 2 chance — and actually hurt our chances with those seemingly simple assumptions.
Being able to reconsider your assumptions is a major tool in the puzzler’s solving kit. Plenty of tricky crossword clues depend on you associating the clue with one thing, when the answer is something quite different.
After all, if you saw the clue “Unlocked” for a four-letter entry, you’d probably try OPEN before you tried BALD. Clever constructors are counting on that.
So be sure to remember Monty Hall and his three-door conundrum the next time you’re stumped on a puzzle. Maybe the answer is as simple as trying another door.
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