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What is the Monty Hall Problem?

The Monty Hall Problem is a probability puzzle named after the host of US gaming show Let’s Make A Deal which was popular in the United States during the 60s and 70s.

One of the show’s biggest segments was the ‘Big Deal’ which saw the show offer two contestants a chance of winning a large prize by selecting one correct door out of three.

The segment has changed somewhat since the show’s original run as it now involves a single contestant picking one of the three doors. Behind one of the doors is a high-quality prize such as a car while the other two doors hide non-prizes such as a goat or nothing at all.

After the contestant selects a door in the modern-day version of the show, the host would then open one of the other two doors, revealing it to be a “loser” door with a non-prize or nothing at all. The host would then ask the contestant whether they would like to switch their original selection to the door he left unopened or stay with their original choice.

Does it Matter if You Switch?

Now, if you were the contestant on the game show, you’d probably think that the host was attempting to trick you into switching doors to the non-prize, and that’s where most people go wrong. You’re actually more likely to win by switching doors.

While the problem has left many people confused, including professional mathematicians and people with PhDs, studies have actually revealed that switching your door doubles your chances of winning.

But why? Well, it’s all about information and probability.

Marilyn Vos Savant’s explanation

Parade columnist Marilyn Vos Savant popularised the problem when she provided an explanation in a 1990s column for the popular magazine. In the explanation, Savant revealed that the solution to the problem is all based on statistics, probability and information.

However, her explanation was hit with criticism from professional mathematicians and people with PhDs. To this day, the problem still leaves people confused.

So How Does it Work?

Some people find it hard to get their head around this, but there’s an easy way of understanding it all.

Now, there are two things you have to remember here. Firstly, the host can only show you non-prizes and can never show you the actual prize. Secondly, you have to remember that there are two non-prizes and only one prize.

With this in mind, your first choice was a complete gamble and, because you understand that the host knows which doors are “losers”, it’s more likely that the door you first selected was a goat or non-prize because there are two of them.

So it’s Better to Switch?

Believe it or not, the host can help you by filtering out the “loser” door and leaving one door unopened; a champion door.

Here’s an exaggerated example: imagine there were one hundred doors and after you’ve chosen your door, the host opens the other 98 all hiding non-prizes. This leaves you with two doors, the one you had originally selected and a special door that the host has left unopened.

You’d feel more confident opening the special door the host decided was better than the other 98 doors, right? You can probably now see how the host’s action actually helps the player. He’s letting you choose between a “champion” door and a generic door. So which one sounds better?

The Probability

Another explanation of this is that when you first selected a door, the probability of winning a car was ⅓ or 33.3% because it was a completely random choice and you had no information at the time. In fact, you probably have a higher chance of selecting a “loser” door at the beginning because there are two of them than selecting the winning door.

However, after the host opened the second door, or the rest of the “loser” doors, the chance of winning by switching doubles to ⅔. This is because you now have more information and the probability of the two doors you didn’t originally select come together.

What happens if the contestant selects the correct door first try?

It’s unlikely that the contestant would actually select the correct door on their first try, but it can happen. If it does, the entire game changes.

For example, if you select the winning prize-door, the host is left with the two loser doors. He would then have to reveal that one of the two doors is a non-prize and randomly pick one to show off. Now, switching doors at this point in the game would mean you’d lose the prize, but sticking with your original choice would mean you’d win.

Still, it’s far more likely that you’ll win by switching doors than by staying with your original choice. This is because there’s a 33.3% chance of selecting the correct door at the beginning, but the likelihood of winning doubles to 66.7% if you decide to switch doors. So you’re far more likely to win by switching doors.

The Monty Hall Problem even appeared in a movie

The problem is so famous that it even appeared in 2008’s 21, an American heist film directed by Robert Luketic. In the film, MIT professor Micky Rosa (Kevin Spacey) challenges student Ben Campbell (Jim Sturgess) with the Monty Hall problem.

After Kevin Spacey’s character asks his student to select a door and reveals the other “loser” door, Ben states that when he was originally asked to select a door he had a 33.3% chance of choosing the correct door.

However, when a second door is opened, Ben states he now has a 66.7% chance of winning because the host had filtered out the “loser” door.