Would you switch?

[forex_chartist]

Quote from ivanbaj:

No switch

1/3 x 1/2 = 1/6 you have to be right in both of your choices

Switch

2/3 x 1/2 = 2/6 yor are saying screw the first choce, let me switch, this replaces the probability of the first step to the opposite of the 1/3 which is 2/3, you still have to be correct on the second choice which brings the total to 2/6

You must know that the sum of all probabilities for a given choice must equal 1

So

P for the first choice is 0.33 the opposite of this choice is 1- p or 0.66 or 2/3 this is why deciding to switch, replaces 1/3 with 2/3

The above assumes that the host knew where the winer is. This is why switching replaces 1/3 with 2/3, him opening the door is not a guess. He is helping you by reducing the options.


If the the host didn't know where the winer is the equation changes (his choice is a guess and it must be included into the total probability calculation)

1/3 for your first choice multiplied by 1/2 for the host choice multiplied by 1/2 for your second choice regardless if you choose to switch or not. So it doesn't matter if you decide to switch or not it is 50:50 at that point.

Here is the answer and explanation of the main actor for this question in the movie 21:


Micky Rosa: He says, "Ben, do you want to stay with door number one or go with door number two?". Now, is it in your interest to switch your choice?
Ben Campbell: Yeah.
Micky Rosa: Well wait, the host knows where the car is. So how do you know he's not trying to play a trick on you - trying to use reverse psychology to get you to pick a goat?
Ben Campbell: Well I wouldn't really care. I mean, my answer's based on statistics - based on variable change.
Micky Rosa: Variable change? But he just asked you a simple question.
Ben Campbell: Yeah, which changed everything.
Micky Rosa: Enlighten us.
Ben Campbell: Well, when I was originally asked to pick a door, I had a 33.3% chance of choosing right. But after he opens one of the doors and re-offers me the choice, it's now 66.7% if I choose to switch... So yeah, I'll take door number two and thank you for the extra 33.3%.
Micky Rosa: Exactly. People, remember - if you don't know which door to open, always account for variable change.
Micky Rosa: Now see, most people wouldn't take the switch, out paranoia, fear, emotions. But Mr. Campbell, he kept his emotions aside and let simple math
[in a game show host voice]
Micky Rosa: get his ass into a brand new car!... Which is better than that goat you've been driving around campus...

 

[intradaybill]

Quote from Scataphagos:

No rational basis for either holding what you have or switching. Odds of ending up with the winner are 50:50, regardless.

Wrong, I mean dead wrong. This is the Monty Hall problem. YOur chances to win are increased to 2/3 by switching.

http://en.wikipedia.org/wiki/Monty_Hall_problem

 

[intradaybill]

Quote from Scataphagos:

Could be.. but I got As in Calculus and was a Summa Cum Laude* graduate in chemistry...

Maybe it was a Scata Cum Laude

http://www.urbandictionary.com/define.php?term=scata

 

[orange_trad]

Quote from traderich:

Remember the old 'Let's make a deal show'?

Here is the scenario:

- You have 3 doors to choose from. You pick one door.
- One is a winner, the other 2 are losers.

You choose a door and one of the other 2 doors is revealed to show its a loser.

Question: Now that you have seen one of the other 2 doors has been revealed to show it was a loser, would you now switch your original choose of a door to the other remaining unopened door and WHY?

Did you just watch 21?

 

[Pekelo]

Marylin had a correct, but not so obvious answer to this Game Show problem. About half of her readers didn't believe she was right and it started a very long discussion.(just like here)

http://www.marilynvossavant.com/articles/gameshow.html

Bottom line, you should switch....

For extra credit, the explanation why you should switch:

"The "subtlety" missed by the mathematicians earlier which they now realize is that the game show host doesnt simply pick "door #2" , so the host has no chance of picking the auto. He has extra info so he's not equally likely to pick any of the remaining doors."

 

[traderich]

The correct answer is you should always SWITCH! Your odds improve from 33.3% to 66.66%. The answer has been explained in the posts.

I am very impressed with some of the folks on this site. Some very intelligent folks.

Also, yes the person showing the other door does know he is revealing a losing door. So, he/she will always be showing a losing door as only one door is a winner.

p.s. I have a BSME and almost finished my MBA. I had a probrabilty/stats course and aced (100 score on every exam and final). I love this stuff!

And, I was going to mention the "Let's make a deal" scenario in which you pick one of 24 cases and get down to 2 suitcases. One is say $250,000 and one is $10. Would you switch. Answer is of course ABSOLUTELY! Notice how in the new "Let's make a deal" with Drew Carey the contestant is not allowed to switch their door.

Anyone read my post on Blackjack? Interesting how you can beat the casino. Well documented and proven yet so many folks (donkeys) just give their money to the casinos. Just having a 0.8% edge on the house can earn you $35-$70/hour. That is, until they figure out that you are, as they called me an "advantaged" player! Fuck the casinos!

 

[Pekelo]

Quote from traderich:

I have a BSME and almost finished my MBA. I had a probrabilty/stats course

I have Google, and I posted the same and correct answer like yours FASTER than you.

 

[virtualmoney]

Given a fixed time period,say 1 week, the stock market can go up,down or stay sideways.
A hedge fund can long, short or use rangebound(options strategy) for that week.

Hedge fund manager A opened long position the market on monday.
On tuesday, there is a major news/economic data that can be interpreted either way so the market will breakout of the range to move in one unknown direction(3rd door is out).

Hedge fund manager B comes along after the news. Should he
(1) randomly pick one direction?
(2) instead of analysing market direction, simply bet that fund manager A is a loser?

 

[Bolts]

Quote from intradaybill:

Wrong, I mean dead wrong. This is the Monty Hall problem. YOur chances to win are increased to 2/3 by switching.

http://en.wikipedia.org/wiki/Monty_Hall_problem

I don't get where the "3" comes from.

You choose door #1...

Once a goat has been revealed behind door #2, there are only two possibilities. car,goat,goat or goat,goat,car.

If the goat is revealed behind door #3, again there are only two possibilities. car,goat,goat or goat,car,goat.

 

[Bolts]

Quote from Bolts:

I don't get where the "3" comes from.

You choose door #1...

Once a goat has been revealed behind door #2, there are only two possibilities. car,goat,goat or goat,goat,car.

If the goat is revealed behind door #3, again there are only two possibilities. car,goat,goat or goat,car,goat.

Ah, never mind. I remember what's wrong with this logic. Just because there are two possible outcomes doesn't necessarily mean they have equal probabilities. It's kind of like flipping a coin that's heavier on one side. When the host reveals a goat, he's not just reducing the number of possibilities, he's also giving you more information about the probability of each of those possibilities, so that 50/50 no longer applies.