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# Interesting Maths puzzle watch

1. My friend and I were arguing over this the other day. He had read a book on it where this woman had proved many mathematicians wrong! Here is the situation:

Imagine you are on a gameshow. There are 3 doors. 1 of those doors has a car in it (the other 2 have sheep!). You get to pick 1 door. You pick your door but you do not open it. Once you have picked, the gameshow host will then show you one door with sheep (different to the one you chose). You then have the option of sticking with your door or changing to the other door.

My friend was saying:
The probability of winning the car if you stick with your door is 1/3
The probability of winning the car if you change the door is 2/3.
REASONING:
The only way you could win if you stick with your door is if you chose the door behind which the car is in the first place, and so your chance of winning is only 1/3.
The other door therefore has 2/3 chance of winning.

However, I was saying:
The probability of winning the car if you stick with your door is 1/2
The probability of winning the car if you change the door is 1/2.
REASONING:
Once shown the door with the sheep in, a new situation is formed, with one car and one sheep, and therefore 50/50 chance.

BTW I completely understand what my friend was saying and his reasoning, and accepted it mathematically. However, I would honestly tend to think of it as a new situation once 1 sheep is shown. Overall, we are both right, but I guess my friend is more right.
It really depends on how you think I guess.
2. Monty Hall Game Show problem, yes, very famous that one!

Probability is very difficult to understand properly at A-level. It wasn't until (as late as) the 30's when Kolmogorov put probability on a rigorous footing...and it's apparently way above my head somewhere at second and third year undergraduate level.

I confuse myself all the time by thinking about probability in too much detail.
3. (Original post by Spenceman_)
Monty Hall Game Show problem, yes, very famous that one!

Probability is very difficult to understand properly at A-level. It wasn't until (as late as) the 30's when Kolmogorov put probability on a rigorous footing...and it's apparently way above my head somewhere at second and third year undergraduate level.

I confuse myself all the time by thinking about probability in too much detail.
well i dont do stats, so probability at A-level standard isnt for me
however even i can get caught up thinking about probabilities.
4. ...your friend is wrong and you are right !!

The chances of having picked the right door , after the sheep one is shown , is 1 in 2.
5. (Original post by jdlover)
...your friend is wrong and you are right !!

The chances of having picked the right door , after the sheep one is shown , is 1 in 2.
lol, depends how you look at it. as i said before, the reasoning behind the 1/3, 2/3 chances is taht the only way you could win the car by sticking with the door is if you chose the door where the car is behind in the first place (before any revealing) and therefore this probability is only 1/3. hence changing is 2/3 chance of winning. still debatable though.
6. imagine 1 million doors you pick one. host removes all the other doors except two, yours and another one. which one would you go for now? is it likely you picked the right one to begin with?
7. (Original post by DaveManUK)
imagine 1 million doors you pick one. host removes all the other doors except two, yours and another one. which one would you go for now? is it likely you picked the right one to begin with?
Good way of illustrating it!
8. (Original post by Jump)
Good way of illustrating it!
thanks there are lots of paradigms for this problem since it was solved but that was my favourite.
9. (Original post by DaveManUK)
thanks there are lots of paradigms for this problem since it was solved but that was my favourite.
Sadly I remember seeing this problem on one of those christmas lectures they used to show when I was about 8-9, totally forgot about it till now!

Tis a good one though, fools most people first time round.
10. Rides by on Jesusaurus Rex. This is old.
I often have to convert the non believers on this one . It is better to change. I demonstrate it with about 10 playing cards face down. I ask them to pick the ace of spades or something. Then I turn over all the other cards except the one they picked and the ace of spades. I once had one gorm who was so convinced he was right about it being 50/50 he stayed with his initial choice every go. He didn't win too many .
11. (Original post by DaveManUK)
imagine 1 million doors you pick one. host removes all the other doors except two, yours and another one. which one would you go for now? is it likely you picked the right one to begin with?
What if he just opens one door and there are still 999 999 doors to choose from?

http://en.wikipedia.org/wiki/Monty_Hall_problem
12. (Original post by Gaz031)
What if he just opens one door and there are still 999 999 doors to choose from?

http://en.wikipedia.org/wiki/Monty_Hall_problem
Change. The one you chose has a 1/1,000,000 chance of winning. Any one of the other unopened doors has a 999,999/999,998,000,000 ≈ 1/999999 chance of winning.
13. (Original post by SsEe)
Change. The one you chose has a 1/1,000,000 chance of winning. Any one of the other unopened doors has a 999,999/999,998,000,000 ≈ 1/999999 chance of winning.
Hmmm. But he was always going to open one door anyway, so did you really pick from 1 million or from 999,999?
14. (Original post by Gaz031)
Hmmm. But he was always going to open one door anyway, so did you really pick from 1 million or from 999,999?
You don't know which one he will pick.
15. ok there are a million doors, you pick one, the host removes one door, which isnt a winner. your chance of winning before was 1/million. now there is a chance of 1/999,999 in winning, by choosing an arbritary door other than the one you have chosen. by changing you increase your chance of winning. is that okay?
16. (Original post by DaveManUK)
ok there are a million doors, you pick one, the host removes one door, which isnt a winner. your chance of winning before was 1/million. now there is a chance of 1/999,999 in winning, by choosing an arbritary door other than the one you have chosen. by changing you increase your chance of winning. is that okay?
Yep. New chance of winning is 1/999,998.999998999998999998....
17. Nooo tis a new situation it's gotta be 1/2 and 1/2 ...to say it's 2/3 would just be incorrectly adding up fractions, since there's only two youd need a common denominator (2), and since there's 2 of them it must be 1.. so 1/2 and 1/2...
18. (Original post by HearTheThunder)
Nooo tis a new situation it's gotta be 1/2 and 1/2 ...to say it's 2/3 would just be incorrectly adding up fractions, since there's only two youd need a common denominator (2), and since there's 2 of them it must be 1.. so 1/2 and 1/2...
When you pick your door, there is a 2/3 Chance it is one of the others.
Even when one is opened, the probability is still 2/3 that the remaining door contains the "prize".

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