I was watching Brooklyn 99 Season 4 Episode 8 around the 5 minute mark
The problem goes “There are 3 doors behind one of which is a car. You pick a door and the host, who knows where the car is, opens a different door showing nothing behind it. He asks if you want to change your answer.
Apparently the math dictates that you have better chances if you change your decision. Why? 2 doors 50/50 chance, no?
One character (Kevin) says it’s 2/3 if you switch 1/3 if you don’t. What? How? Please help.
In: Mathematics
The reason is quite simple.
If your initial choice was incorrect, and you switch, you have a 100% chance of getting the car.
Since the odds of your initial choice being incorrect were 2/3rds, therefore the odds of winning if you switch is *also* 2/3rds.
One way to solve it is to list all the scenarios:
Assume car is behind door #1.
Car | You choose | Monty opens | You switch | Don’t switch
———|———-|———-|———-|———-
1 | 1 | either 2 or 3 | lose | win
1 | 2 | 3 | win | lose
1 | 3 | 2 | win | lose
Doors 2 and 3 are the same.
If you don’t switch, your odds are always one in three of being correct.
Now break down the odds when you switch:
There is a one in three chance you picked the right door originally and are losing it by switching.
There is a two in three chance you picked the wrong door originally, and since Monty eliminated the other incorrect door you are switching to the correct one.
Simple.
I still get confused by it, but the simplest way I can explain is that, when you pick a door, the chances you picked an empty door (no prize) is 2/3. The fact that the host reveals a door doesn’t change much. You still know there is a 2/3 chance that you picked the wrong door on your first choice and a 1/3 chance that you picked the right door. So when he asks if you want to switch, he’s really showing you that there were really always only 2 options: opening the 1 of 2 incorrect doors you most likely picked or picking the 1 door left that the host knows is correct.
_____________________________________________________________
|___ **your door** (which is most likely empty) ___|
______________________________________________________________
|___**the last unopened door** (that is most likely correct)___|
hope this helped
Which is more likely?
You picked the one door out of three that has the prize.
Or;
You picked one of the two doors that does not have the prize.
Chances are that you picked one of the two doors that does not have the prize, it is a 2/3 chance that you chose wrong. It is a 1/3 chance that you chose the one with the prize.
Since it is more likely that your choice was wrong, it would be wise to change your choice.
On your first choice you have a 33.3% of being right. But you also have a 66.6% of being wrong.
The opening of another door does not change the fact you only had a 33.3% of choosing the right door in the first place.
But with one door opened the full 66% chance of you being wrong “collapses” to the other closed door you didn’t choose.
Imagine there was 1,000,000,000 doors. You pick one. The host then opens up 999,999,998 doors. And asks if you want to switch. You clearly did not have a 50:50 chance of picking the right door even though there’s only two closed doors out of a billion. You had only a 1 in a billion chance of picking the correct door. And a 999,999,999:1billion odds you chose wrong.
It’s easier to explain backwards. When you pick one option out of three there is a 2/3 chance that the correct answer was behind the other doors. You picked one there were two left. The odds say that one of the ones you didn’t pick has the prize. If you could pick 2 doors instead of one you would have a 2/3 shot of winning. However if you pick 2 doors you know one has to be empty. In this scenario it is like picking 2 doors and the host tells you which one is empty and then let’s you have the last one.
Let’s do the experiment:
Take 3 cups and 1 ping pong ball. Line the cups up in a row. Place the ball under cup 2. You switch cups every time offered by the host.
First experiment you pick cup 1. The ball is under cup 2. The host shows that cup 3 is empty and asks if you want to switch to cup 2. You do and win.
Second experiment you pick cup 2. The ball was under this cup. Host shows you cup 3 is empty and asks if you want to switch to cup 1
You do and lose.
Third experiment you pick cup 3. The ball is under cup 2. Host shows you cup 1 is empty and asks if you want to switch to cup 2. You do and win.
You win 2 out of 3.
The key thing is that the host knows which door has the car.
Imagine a more extreme case. There are 100 doors with a car behind 1 of them. You pick door 7. The host reveals the goat behind every door except door 7 and door 38. Suddenly, door 38 looks suspiciously appealing…
You had a 1% chance of getting the right door. This means there’s a 99% chance that door 38 is the right one.
Again, the trick is that the host knows the right door. If the host didn’t, then most of the time the host would accidentally reveal where the car was. It’s the *host’s* knowledge that changes the odds.
Posting this as a top level comment in case this explanation helps anyone;
The break down in logic most people have is not realizing the way the initial condition is set up. Look at it from the first pick.
You have a 2/3 chance of picking the wrong door initially, so that is most likely what you do. Now after you pick he eliminates one of the wrong options, but you already picked based on 2/3 probability, so you likely already picked the wrong door, thus meaning that if you were to switch doors 100% of the time after given the option you will switch from a wrong door to a right door 2/3 of the time.
The explanation that clicked for me was to think of them as two groups. Group A is the door you choose. Group B is the other two.
After your first choice, there’s a 33% chance it’s in A and a 66% chance it’s in B.
Now what if they said, “Do you want to stick with A, or do you want to switch to B?” Of course you’d switch, because group B has two chances.
When they remove one of the bad ones, there’s still a 66% chance it’s in group B. By letting you switch after getting rid of one of the losers, they’re still giving you *all* of Group B.
It’s kind of like they let you pick two doors instead of one.
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