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Monty Hall Prob

2009-02-24T05:42:00.000-08:00

Monty Hall problem

(Just in case you guys were wondering what the lecturer was talking about during the probability lecture. This qn is also featured in the book "The curious incident of the Dog in the Night Time", so those aquainted with the book will have come across this qn)

A simple Tree diagram to illustrate:

Different ways of arriving at the Answer:

Increasing the number of doors

It may be easier to appreciate the solution by considering the same problem with 1,000,000 doors instead of just three. In this case there are 999,999 doors with goats behind them and one door with a prize. The player picks a door. The game host then opens 999,998 of the other doors revealing 999,998 goats—imagine the host starting with the first door and going down a line of 1,000,000 doors, opening each one, skipping over only the player's door and one other door. The host then offers the player the chance to switch to the only other unopened door. On average, in 999,999 out of 1,000,000 times the other door will contain the prize, as 999,999 out of 1,000,000 times the player first picked a door with a goat. A rational player should switch. Intuitively speaking, the player should ask how likely is it, that given a million doors, he or she managed to pick the right one.

Combining doors

Instead of one door being opened and shown to be a losing door, an equivalent action is to combine the two unchosen doors into one since the player cannot, and will not, choose the opened door (Adams 1990; Devlin 2003; Williams 2004; Stibel et al., 2008). The player therefore has the choice of either sticking with the original choice of door with a 1/3 chance of winning the car, or choosing the sum of the contents of the two other doors with a 2/3 chance as shown.

The game assumptions play a role here—switching is equivalent to taking the combined contents if and only if the game host knows what is behind the doors, must open a door with a goat, and chooses between two losing doors randomly with equal probabilities.

The only difference between trading for both doors and the trade that is actually offered is whether the host opens one of the two doors. Opening one shows which of these doors the car must be behind if it is behind either. At least one of the two unpicked doors contains a goat, and the host is equally likely to open either of these doors so opening one gives the player no additional information; opening one does not change the 2/3 probability that the car is behind one of them (Devlin 2003).

And of course the most Important:

Why the probability is not 1/2

The most commonly voiced objection to the solution is that the past can be ignored when assessing the probability—that it is irrelevant which doors the player initially picks and the host opens. However, in the problem as originally presented, the player's initial choice does influence the host's available choices subsequently.

This difference can be demonstrated by contrasting the original problem with a variation that appeared in vos Savant's column in November 2006. In this version, Monty Hall forgets which door hides the car. He opens one of the doors at random and is relieved when a goat is revealed. Asked whether the contestant should switch, vos Savant correctly replied, "If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch" (vos Savant, 2006).

In this version of the puzzle, the player has an equal chance of winning whether switching or not. Assuming the player picks Door 1 there are six possible outcomes that can occur, each with probability 1/6:

	Player picks Door 1
	Car behind Door 1		Car behind Door 2		Car behind Door 3
Host opens:	Door 2	Door 3	Door 2	Door 3	Door 2	Door 3
Host reveals:	Goat	Goat	Car	Goat	Goat	Car
Switching:	loses	loses	?	wins	wins	?

In two cases above, the host reveals the car. What might happen in these cases is unknown—perhaps the contestant immediately wins or immediately loses. However, in the problem as stated, the host has revealed a goat, so only four of the six cases remain possible, and they are equally likely. In two of these four cases, switching results in a win, and in the other two, switching results in a loss. Staying with the original pick gives the same odds: a loss in two cases and a win in two others.

The player's probability of winning by switching increases to 2/3 in the original problem because in the two cases above where the host would reveal the car, he is forced to reveal the remaining goat instead. In the table below, these two cases are highlighted:

	Player picks Door 1
	Car behind Door 1		Car behind Door 2		Car behind Door 3
Host opens:	Door 2	Door 3	Door 3	Door 3	Door 2	Door 2
Host reveals:	Goat	Goat	Goat	Goat	Goat	Goat
Switching:	loses	loses	wins	wins	wins	wins

This change in the host's behavior causes the car to be twice as likely to be behind the "third door", and is what makes switching twice as likely to win in the "host knows" variation of the problem.

*******

Understand? I think "Why the probability is not 1/2", is the biggest explaination there is, it's quite clear up there, if you bother to read everything:P

Oh wells, good luck in future lectures:D:D:D

Disclaimer: I copied and pasted from Wiki

ATTENTION OUTING!

2009-02-24T05:34:00.000-08:00

Wonder why my econs notes so clean... cos I fell asleep during lecture. Hahas...

Anyway, anyone else beside yours truly and Dennis who is running for Council?

*****

Next monday are you guys free? Come early and we can have breakfast. Perhaps take a ride down to Upp Thomson area to eat Roti:) Just a break from all the hard work. (So please MUG over the weekends so you dun feel guilty tt you didnt mug on mon morning:D) I'll ask if Jon, Ame, Gab and TZ will be able to make it down too. Just nua there and chill for abit before returning to school to begin lectures and stuff.

Hope that you guys can make it! (Upp Thomson has alot of good food:D *hint* *hint*)

All the best in doing your tutorials and catching up on some late sleep:D If dno how to do then bring on mon or sth, can ask a few geniuses in our grp:)

OH WAIT!

ONE MORE THING!!!!

buai<3

2009-02-21T06:42:00.000-08:00

Hi I'm very sad because my phone is screwed and i wanted to sms yall to ask if a monday breakfast was possible but now i can't D:

and somehow facebook's not working for me either, and my macbook battery is trying to be funny as well. -sadface

-teezee

HELLO BW08<3

2009-02-19T07:10:00.000-08:00

hi this is a very very very bored+sleepy kristabelle
who cannot sleep because she's doing ISLE shitty response thing
i hate schoooool gahhhh
anyway
see you pple tmr at bwalentines
im so sorry i always miss og outings,
but im going for tmr's one :D
I LOVE Y'ALL
(even alimi and avinash who are such meanies, esp alimi)

love,
me :)

ps: this font so cool yo! B)

(C)

2009-02-18T03:16:00.000-08:00

Loser lo lo 10S03R

loser lo lo 10S03R

XDXD

***

Felt so tired when I went to school today. Seriously. Soearlylaomgwodetian. Slept during econs lecture. NOT GOOD! Gp was interesting cos we had a debate: "Should we allow females so serve in NS so as to maximise our manpower and better defend our country?" I insist that my side had a better argument cos the proposition kept going out of point! Harping on "yes we should pump all our females into the army cos more people = more military strength". Please la! And I have absolutely no idea why we lost, like seriously. Kelong one. Pfffbt.

Anyways, had training later. Sian, strokes all WRONG WTH!!! Really made me sian to the max (sttm) lor. Saded. And physical made me realise how much my stamina and stuff has degraded. 3km run on Tues alr run so slow lo. Sian.

Today is a sian day. And I threw a ball which accidentally hit Shimin on the head.

But I met Aloy on the way back afr trg (and Angus too), and went back with him tgt on the bus. At least some nice part of the day, just talking abt stuff all the way back. Pretty cool.

I'm sorry, 3rd post on the blog and I'm droning off liddat le. HAHAS!!! I should just write this somewhere else.

Btw right, JonLim has a blog leh. Can go check it out if u want:

www.point-v.blogspot.com

Enjoy the rest of your day you guys!

***

A quote from JL's blog:

'Every song has a CODA, a final movement. Whether it fades out, or crashes away. Every song ends. Is that any reason not to enjoy the music? The truth is, there is nothing to be afraid of. It's just life.'

- One Tree Hill

#2. XOXO

2009-02-18T02:53:00.000-08:00

YOU KNOW YOU LOVE ME XOXO

you should know by now who this is rightt. IF NOT, you seriously dont knoww me.

hello school sucks my timetable sucks im v.tired of lectures and lab sessions school is v.annoying.

thats all ihave to say now.

thanks cherise for setting up the blog(:

Welcome message

2009-02-18T02:14:00.000-08:00

Welcome one and all to BW08 '09's blog! This is our very first common website, where all members of bw08 as well as our lovely OGLs can post their thoughts, daily events, feelings and rants. For those who are frequent bloggers, no need further explainations on how to use the blog and what to write. For those who have never blogged or have never considered blogging, HELLO! It's time to start. It may feel abit unfamiliar for the 1st time, and you might feel a little lost on what to write, it's ok. It starts out like that. But you'll get the hang of it after a few practises. It's really good practise for learning how to open up about your thoughts and feelings, and of course using good english will help in GP too:D After all, you're writing an essay each time you post.

Post about anything! It's fun after awhile, really. My hopes is that this blog will link us all together wherever we are, and will be brimming with activity (or as JonLim says: "Very HAPPENING"). Yea, vocab abit limited... :P Hm, post about your day, the occasional frivolous talk, gossips, rant here (but please keep a limit to the vulgarities. I trust that there is a need o minimise it, then again if it's that bad and no other word can replace it then by all means. We'll understand:)). And if you're really up to it, perhaps your insights and thoughts on certain topics or events occuring around you (like a deep post). It will help the other members to gain an insight to your personality, and help us to understand you better. We all want people that can accept us for who we are, so by opening up and sharing your thoughts and feelings, it will really help us and you to get to know and accept one another better:D:D:D

SO! I'll invite the rest of the OGmates by emailing them the user and pw (once I've explored the settings and realise how to do so), so you guys can do posts too! I need your help maintaining the blog too:D Oh yes, any events (eg. OG outing, dinner, House Party, birthdays, tcss-sessions, mugging tgt days, match supports, chilling out etc) will also be posted here so please keep a lookout!

Ok, now on to the other stuff. I need help with the background. People who are familiar with Photoshop or photo editing, I need your talents! I'm thinking of creating a more personalized skin for this blog, something that will be closer to us, rather than hook something off from blogskins.com etc. My idea is something that will capture all of us, like a photo, abit of editing here and there to make it nicer and stuff, featuring our OG as well as OGLs. I don't have this skill, so I'm asking around for help. Or if you know anyone else that's so kind to help us then that's fine too:) You have my deepest thanks!

OKAAAAYYY~! I'm so on on this project. Hahas, well, hope you guys will enjoy this as much as I do! We'll meet in school again! TATA!!

All the best wishes,

(C)herise Cho~