The Monty Hall Problem - And an explanation that doesn't involve trials

The Monty Hall problem is as follows:

Monty Hall gives you the choice of picking from three doors: A,B, and C.
Two doors have a goat behind them - the other door has a car.
You pick a door. (A for example)
Monty then reveals that behind a different door is a goat (B for example).
You are given the choice of switching the door you had picked earlier (to C, for example).
Should you switch to door C, or stick with door A?

At first, it may seem, as if it doesn't matter. You have a 50/50 chance between A and C. But, this is not the case. If you actually perform this test with another person you will see that if you force yourself to switch doors every time, you will win the car 66% of the time.

Searching online, I found all kinds of websites describing this problem. When I tried to find out why this happened though, I came up short. Most websites said "when you test it, it works." This is not a good enough explanation. It's as if there is a lightning machine, where everytime you pull the lever, you get struck by lightning - and rather than figuring out why, you just test it over and over again. So, I will not be accepting "Because the trials say so" for the question of "Why does it work?" - I thought about it for a very long time, and have come up with this answer which hopefully should be simple, and does not at all require trials.



I'm inserting door letters to make the example a little easier to follow, but they should not be necessary.
You make your first choice (Door A). This door is very likely wrong - In fact, the chances of it being wrong are 66% percent. Ouch :(
Monty shows you that a different door (Door B) is wrong - This door has a 100% chance of being wrong.

(If you are interested in seeing the similarities between this problem and the Deck of Cards example below, it may be interesting to note here that ALL choices except for your original choice, and Monty's remaining door are gone)

You now have the opportunity to switch - you can pick between Door A and B.
At the beginning, didn't we say A had a 66% chance of being wrong?
And when we started, together B and C had a 33% chance of being wrong. Now that C is eliminated, B retains that original 33% chance of being wrong.
Between A and B, B (the switch) has a better chance of winning

Of course you may wonder, "don't the probabilities change once he opens the door?" - The only probabilities that changes when he opens the door is the chance that the door he will open (door B in the above example) is wrong - it goes from 66% to 100%, and the probability of B and C being correct shifts its 66% chance of winning over to only Door B . Everything else remains the same the entire time


Here is a much easier example to understand, which led me to my description above.
Imagine instead of a door, Monty gave you a Deck of Cards, and said that if you pick out the Ace of Spades, you win.
You pick a card - there is a 1 in 52 chance this card will win.
Now, Monty removes ALL cards from the deck, except for one (if the Ace of Spades is not in your hand, it's in Monty's now).

(Remember this part of the 3 Doors example above?)

The card you picked has a 1 in 52 chance of winning....... Wanna trade?
Monty's card must have a 51 in 52 chance of winning. After all, Monty essentially had 51 cards to search through.

This example is alot harder to apply on a smaller scale such as the door example, but it works. Huzzah.


If any clarification is needed, post in the comments, and I'll do my best to help.

Saitek Eclipse II

The Saitek Eclipse II keyboard is a USB 104-key keyboard. One of it's most obvious features is that it is backlit - every key is illuminated in either Red, Blue, or Purple, and some of this light also escapes in between keys and on the edges of the keyboard.

Now, if you were to look at the Eclipse II straight out of the box, you wouldn't notice anything particularly unique about it. The keys are spaced well, and the letters on each key are fairly large (to allow light to shine through). There is a cluster of media keys in the top right corner that control track fwd and rw, play/pause and stop, volume up and down, mute and color. There is also a dial that is used to adjust the brightness.
You'll also notice that the keyboard feels very solid. Without being quite heavy, the keyboard feels very firm, and the glossy plastic that is on it's surface (between the insert/del/home and arrow keys for instance) looks great.

This keyboard's features become evident during use. When first powered up, the blue LED's illuminate beneath the keyboard making every key glow. The action is excellent, with average depth strokes and very good rebound force. The keys are also very quiet - they don't have the standard annoying click click that many keyboards ( including my mom's Dell keyboard, and my old Logitech LX 300 keyboard) have. In a well lit room, the purple setting shows up best - luckily it's a very subtle purple, that matches the black of the keyboard very well. I think I'll be sticking with the blue or red settings for the day-time, switching just for the heck of it - all the colors really are great, and i don't think I'll want to be sticking with just one.

I tested the keyboard out on both an OSX and a Vista environment and didn't have any problems in either situation. In Vista, in order to get the media keys working with iTunes when iTunes is minimized (or not the window with focus), I had to download a plugin called mmkeys.dll from Everything iTunes. I haven't tested any of the media keys in an OSX situation.

Newegg currently has the keyboard for US $50 with $8 shipping.

I purchased this keyboard because I often had trouble seeing the keys late at night when the only light I was working by was my monitor. I would definitely recommend this keyboard. Leave a comment if you have any questions.

$10 off at ProWinds.com

I received this coupon in the mail the other day, good for $10 off any order of $50 or more at prowinds.com. It is only good for the first use, and only until 6/30/08.

PW458231

OSX Mouse Acceleration

I've been tinkering around a bit in OSX lately (10.5) and came across a "feature" that really annoyed the heck out of me: Mouse Acceleration. With this feature, the speed of your mouse cursor increases exponentially with the speed your mouse moves across the mouse pad. Now, maybe I'm stuck back in the days when a mouse was shaped like a pointer finger, but I'm used to my mouse acting as an extension for my arm. And my arm and my finger move at the same speed - so shouldn't the mouse? Unfortunately, there's no setting in system preferences to turn this "feature" off. Luckily, there has been some success hacking the formulas that control the amount of acceleration.

Looking for a free solution, I first tried "MouseFix". Mousefix works for some people, but does not work for me. I'm not sure why (maybe because I'm in Leopard).

After the failure with MouseFix I tried USB Overdrive, and what a difference it made. My mouse control feels so much more normal now. The action is a little choppy - if I needed precision for something such as Photoshop, I'd probably lower my DPI. However, the lack of acceleration definitely supersedes the minor choppiness of my mouse, which is only noticed when moving very slow.

XKCD

XKCD is a webcomic full of great stuff. It's updated every Monday, Wednesday, and Friday.
Check it out here: xkcd.com

(comic shown is number 249)

Illuminations - Joseph Alessi

Illuminations, by Joseph Alessi

Genre: Instrumental/Classical

I purchased this album on Amazon MP3 last week, and it has gradually become the most played album on my iPod, and in my iTunes library. With over 3.75 hours of play time in my iTunes library alone (not including on the iPod) it is at the top of my play counts, and it has only been there one week.

Mr. Alessi is possibly one of the best trombonists I have ever heard. His tone is elegant and natural - if mother nature could make music, this is how she would make it. These views are not shared solely by me apparently:

Joseph Alessi is a world-renowned, primarily classical, trombonist; he is the current Principal Trombone of the New York Philharmonic Orchestra and an active soloist, teacher/clinician and recording artist. Alessi is considered to be one of the finest players of the instrument in modern times.

This album is not Alessi's only accomplishment, though. Having performed on over 15 other CD's, he certainly has a cornucopia of works. I also own Alessi's Trombonastics which is just as technical as it is beautiful. Either album is sure to inspire any serious trombonist.

Ordering the B-Lines Standard Kit Resistance Bands

Last week I ordered the B-Lines Extreme Kit Resistance Bands from BeachBody.com and received them yesterday. I was very happy with the bands, they seem to be very high quality. I wanted some lighter bands though for the exercises that didn't quite require 40-50 lbs of resistance, so I decided to order the Standard Kit of B-Lines. That's where everything went downhill.

First I tried the BeachBody website. After numerous problems with my shopping cart not being recognized and then them not allowing my shipping address, I decided it wasn't worth the effort.

Next I went over to powertobefit.com and tried to place an order. The order went well until I was switched over to Paypal. Apparently my e-mail address has already been registered with Paypal, however I have never done it, and the verification questions are not correct. Now I'm stuck with a purchase order from Power to Be Fit, and no way to pay them. Hopefully they'll cancel my PO.

Finally, I tried Extremely-Fit.com.
Third time's the charm I guess. After about 10 minutes I had an invoice with shipping estimates and the works. It was a very quick and easy transaction (though, I couldn't find any coupon codes). The bands should be arriving at a location where I'll pick them up sometime next week.

I enjoyed my previous transaction with Beachbody.com, but I hope they get their problems sorted out soon. If I have problems ordering with them again, it may be the last time I attempt to place an order.