So, you’ve made it to the last round of a TV gameshow
and have the chance to win a brand new car.
It sits behind one of these three doors,
but the other two have a sad little goat behind them.
You make your choice and the host decides to reveal where one of the goats is.
He then offers you a chance to change your door.
Do you do it?
Does changing your choice even make a difference?
The short answer is:
Even though it seems counterintuitive,
changing you door choice actually doubles your odds of winning the car.
But how is that possible?
This is the “MONTY HALL PROBLEM”.
At the start, most people correctly assume that
you have a 1/3 chance of choosing the correct door.
But it would be incorrect to assume that when one door is removed,
each door now holds a 50/50 chance of having the car.
Let’s use a deck of cards to understand why.
Pick a card from this deck without looking.
This card has a 1/52 chance of being the ace of spades.
但现在 我准备把另外的50张翻过来 剩下一张不翻
But now, I’m going to flip over all the other cards except one,
none of which are the ace of spades
Of the two cards left, which one seems more likely to be the ace of spades?
The one you chose randomly out of a deck of 52
or the one I purposefully and suspiciously left turned down.
It turns out your card remains at a chance of 1/52,
while my card now has 51/52 probability of being the ace of spades.
The same principle is ture with the three doors.
You see, when I removed the door, I did so with motive,
knowing there was a goat behind it,
The only two scenarios that exists are
A, you chose the correct door and I’m arbitrarily picking one of the wrong choices to show you,
in which case staying will make you win;
or B, you pick the wrong door and I show you the other incorrect answer.
in which case switching will make you win.
Scenario A will always happen when you choose the winning door,
and B will always happen when you pick a losing door.
Therefore A will happen 1/3 of the time,
and B will happen 2/3 of the time.
As such switching your door wins two out of the three times.
This paradox has perplexed many people, including scientists and mathematicians to this day,
because our gut tells us that switching will have no consequence.
But when using formal calculations or computer simulators,
the results don’t lie.
Switching your door increases the probability of winning.
Let’s see it one more time using a chart.
Here are all the possible scenarios.
汽车在门1 2 3后 每扇门你都有机会选到
The car is behind door one, two or three, and you have the choice of each three doors.
This means there are nine possible outcomes.
Let’s tell you them up quickly.
如果在门1后 你又选了门1 那你就得保留选择
If it’s behind door one and you chose door one, you should stay.
But if you chose door two or three, you should switch.
如果在门2后 你又选了门2 那你应该保留选择
If it’s behind door two and you chose door two, you should stay,
but the other two you should switch.
Add it all up and you should switch six out of nine times.
So, do you still trust your gut feeling?
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