A contestant on a gameshow is given the choice of three doors.
Behind one door is a speedboat; behind the others, dusty bins.
The contests picks a door.
The host, who knows what's behind the doors, opens another door, which has a dusty bin.
He then says to the contestant, "Do you want to pick the third door or stick with your original choice?"
Is it to the contestant's advantage to switch doors?
Assume the following :-
- The contestant would rather win the speedboard than a dusty bin replica.
- The host is constrained to always open a door with a dusty bin.
- The host is constrained to always make the offer to switch.
posted by steven | 5:30 AM
5 Comments:
Um, this is the Monty Hall problem, right?
In other words, yes, it makes sense to change, every time.
The key question is--if the contestant changes, what is the probability of winning having made the change? (Most people get that probability wrong.)
Although most people get it wrong i have to try. The contestant is better of changing, or at least choose again, because the probability of the speedboat present raised, or rather changed, because a probability of advantage never equals advantage.
No, it's not in the contestant's advantage to change (or not to change). It IS in the contestant's advantage to CHOOSE.
Initial he has a 1/3 chance of getting the boat, right?
After the MC, he now has a 1/2 chance of getting the boat--but it doesn't matter which door he picks. If he CHOOSES to pick #1, it's a 50/50; if he CHOOSES to pick #2, it's also 50/50.
I don't see how his first choice (1/3) is relevant given that the problem has changed.
Yep, it's Monty Hall.
---
The contestant should switch.
The contestant has a 1/3 chance of initially choosing the speedboat and therefore a 2/3 chance of choosing the dusty bin.
If the contestant initially chose the door hiding the speedboat (1/3 chance), switching loses them the big prize.
If the contestant initially chose a bin (2/3 chance), the host must reveal the other bin. Therefore the boat must be behind the remaining door.
Therefore the contestant stands a 2/3 chance of winning the boat if they choose and then switch.
This counter-intuitive result is known as the Monty Hall paradox - more on that here :- http://en.wikipedia.org/wiki/Monty_Hall_problem
This problem was posed by Marilyn vos Savant (who may have the world's highest measured IQ, if you believe in such things) in her 'Ask Marilyn' column :- http://en.wikipedia.org/wiki/Marilyn_vos_Savant.
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