# Intuiting Monty Hall

I have long carried my own intuition for the Monty Hall problem which I have not seen anywhere else. I took the time to write it up for a friend. You might enjoy it too. It is more of an algorithmic, step-by-step approach which emphasises where the critical information is injected into the game that gives us the counter-intuitive result.

# Fully Random Monty Hall

I want you to imagine a different version of Monty Hall, namely one in which the host opens a randomly selected door.

Here’s the scenario: we are a participant in a game show, and we have arrived at a point where there are three doors brought onto the stage. The host explains that behind one of the doors is a big prize! And behind the other two there is nothing.

We’ll get to select a door, which the host will label A. Then the host will randomly pick another door of the two remaining, and label it C. The host is careful to explain that by “randomly pick”, he means that he does not know what is behind any of the doors, so he cannot affect the outcome with his choice.

The final, unchosen, door will be labeled B. We will then be given the choice to switch from door A to door B, or stay with door A. When the host selected door C, it was taken out of the game. To be extra clear: no doors have been opened at this point – they have just been labeled.

In this version of the game, it is perhaps clear that there is no reason to switch from A to B because the prize is still equally likely to be behind any of the doors.

In fact, as we have not yet used any information on where the prize is, we are even at this point free to randomise its location again. In this version of the game, there may have been nothing at all behind the doors when we made our selections – and the location of the prize is determined only once we have made our final choice.

## Prize placement splits the universe into three

That’s probably a good way to look at it! Let’s say that it is only after we’ve made the final choice that the prize is placed behind a randomly selected door. To make sure the door is randomly selected, the stagehands that place the prize are not allowed to look at the front of the doors, which is where the host have labeled which the choices are. Thus, the prize is equally likely to end up behind any of the doors A, B, or C.

At this point, the placement of the prize makes the universe split into three parallel universes, one for each location of the prize.

- In one universe, the prize is behind door A. In this universe, if we stay with A, we get the prize. If we switch to B, we get nothing. Regardless of what we do, the host opens C, revealing nothing.
- In another universe, the prize is behind door B. If we stay with A, we get nothing. If we switch to B, we get the prize. Regardless of what we do, the host opens C, revealing nothing.
- In the third universe, the prize is behind door C. If we stay with A, we get nothing. If we switch to B, we also get nothing. Regardless of what we do, the host opens door C, revealing the prize!

In other words, when this variant of the game is played, three things happen in
parallel universes *if we choose to stay with A*:

- Prize in A (host opens C): we win.
- Prize in B (host opens C): we lose.
- Prize in C (host opens C): the round is declared invalid.

If instead we opted to switch to B, the universe will split into a different set of three universes:

- Prize in A (host opens C): we lose.
- Prize in B (host opens C): we win.
- Prize in C (host opens C): the round is declared invalid.

In this version of the game, it does not matter whether we switch or stay. Whichever choice we make, we will win one third of the time, lose one third of the time, and the round will be declared invalid one third of the time.

# Random Monty Hall With Redirection

The third universe – that where the round is declared invalid regardless of our choice – is unsatisfying though, so we will add a small twist to the procedure to avoid invalid rounds. The game still splits the universe into three. These will be the same in the first two cases, but we make a small modification to the third.

- In one universe, the prize is behind door A. If we stay with A, we get the prize. If we switch to B, we get nothing. Regardless of what we do, the host opens C, revealing nothing.
- In another universe, the prize is behind door B. If we stay with A, we get nothing. If we switch to B, we get the prize. Regardless of what we do, the host opens C, revealing nothing.
- In the third universe, the host is about to open door C with the prize behind it, but someone screams in his earpiece which makes him freeze before opening the door. He understands this means he was just about to invalidate the round, so instead he opens door B, revealing nothing behind it, and says that if we switch, we’ll get door C instead.

If we choose to stay with door A, the outcomes of the three universes are similar to before, except the host has cleverly redirected an invalidation to a loss:

- Prize in A (host opens C, we stay with A): we win.
- Prize in B (host opens C, we stay with A): we lose.
- Prize in C (host forced to open B instead, but we stay with A): we also lose.

Although we no longer have invalid rounds, they were practically losses to us anyway, so if we stay with A we are still one for three in this new variant.

But!! If we switch in this variant, the host’s clever redirection actually forces our hand to select the winning door – the one the host refused to open because doing so would have invalidated the round. This means we get the following set of parallel universes:

- Prize in A (host opens C, we switch to B): we lose.
- Prize in B (host opens C, we switch to B): we win.
- Prize in C (host forced to open B instead, and thus we are forced to switch to C): we win also here!

In other words, if we stay, the host’s redirection makes sure what was about to be a universe with an invalid round becomes a loss. But if we switch, the same redirection causes the universe about to experience an invalid round into a win.

The random prize placement followed by someone screaming in the host’s earpiece is exactly equivalent to the traditional Monty Hall setup, where the host already knows ahead of time which door the prize is behind, and chooses on their own not to open that door.

This is how the game host makes it a 1/3 proposition to stay, and a 2/3 opportunity to switch. If the host opened a door randomly, then 1/3 of the rounds would be invalidated. The host avoids the invalid outcome by specifically choosing a losing door to open, which means they redirect invalidation into a certain win for switchers, and a certain loss for stayers.