The other day Hannah told me about one of the interview questions she was asked about during her interviews with Stryker. The question, come to find out, is a famous one called 'The Monty Hall Problem' or 'The Monty Hall Paradox'.
Here's the question:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
Now.. before you read on.. what would you do? Decided? Okay... read on.
The problem is a probability puzzle based on the TV show Let's Make a Deal and is named after the show's host. The answer appears absurd - but it is demonstraby true.
My well trained engineering brain immediately spit out the answer 'doesn't matter' - which is equivalent to saying 'no' don't switch. You see, with 1 response revealed you have a 50/50 chance of having picked the car. Why switch? Makes sense doesn't it?
Nope: as it turns out the correct answer is to switch!! Hannah informed me.
NOOOOO! I responded. And revved up my statistics and mathematical engine to prove her (or more correctly the person over at Stryker who posed the question) that they were wrong. But I was wrong!
Here's proof that switching is to your advantage:
Say the doors are marked C G G (for car, goat, goat). And I am going to use X to represent the door that the host opens (which always will be a goat door). The door I picked I will mark using a little 'p' (example: Gp means I picked a goat). Here's the three possible outcomes that will happen based on which door you originally pick (1st, 2nd or 3rd):
you pick the first door: Cp G G
host eliminates 2nd door (or third.. doesn't matter): Cp X G
you swap: C X Gp
outcome: LOSE
you pick the 2nd door: C Gp G
host eliminates 3rd door: C Gp X
you swap: Cp (as result of swap) G X
outcome: WIN
you pick the 3rd door: C G Gp
host eliminates 2nd door: C X Gp
you swap: Cp (as result of swap) X G
outcome: WIN
You win 2/3rds of the time as a result of swapping!!
Just to prove to you that staying results in a worse outcome... here's the result for staying:
you pick the first door: Cp G G
host eliminates 2nd door: Cp X G
you keep your original pick
outcome: WIN
you pick the 2nd door: C Gp G
host eliminates 3rddoor: C Gp X
you keep your original pick
outcome: LOSE
you pick the 3rd door: C G Gp
host eliminates 2nd door: C X Gp
you keep your original pick
outcome: LOSE
So - in this case you win 1/3rd of the time. Summarizing, swapping results in 2/3rds chance of winning; staying results in 1/3rd chance of winning. You should swap!
Amazing. Hannah... I never did ask you what answer you provided during your interview.. but swapping is clearly the right answer. I stand humbled and amused.
Happily, I am in good company. Here's a quotation from someone who wrote a famous book about math puzzles: "... no other statistical puzzle comes so close to fooling all the people all the time" and "that even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer." That's pretty close to how I initially responded to Hannah :)
At least old dogs CAN learn new tricks.
Oh, and note the answer does not depend in any way on where you stash the car. It would work out exactly the same for any placement.
Hip History....
7 years ago
7 comments:
Makes sense: you had twice as many opportunities to choose the losing door to start with, so you have twice the chance of switching to the winning door.
It's interesting that a lot of people immediately go to thinking it's a 50/50 chance once the host eliminates one of the doors, but there are still three doors, so you're still dealing in thirds, so it's impossible for it to be a 50/50 chance.
as it turns out - thinking about what your chances of picking the car after one of the wrong answers has been revealed is how you go wrong. in fact, if you had not originally picked an answer and one choice was eliminated right off the bat (before you made a choice)... you would have a 50/50 percent chance remaining of being right.
the trick is that you are not being asked what the probability is that out of the remaining two answers you picked the correct answer (which IS 50/50). you are being asked should you switch.
update: I tried this out on about 10 of my engineering colleagues over the last 24 hours or so. 10 out of 10 said either "stay" or "it doesn't matter".
this is a fascinating paradox!!
i am surprised hannah has not taken me to task via comment. interesting... are you alive gorgeous??
Todd, the answer will be found in this link. Check it out! Or you can refer to the Numbers episode regarding the three armored cars entitled "Money or Nothing"
.http://en.wikipedia.org/wiki/Monty_Hall_problem
Happy Hunting!!
thanks becky!!
and i am loving it that you watch NUMB3RS ;)
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