Cars vs. Goats

There was an episode of NUMB3RS (113 - Man Hunt) that included a brief explanation of the Monty Hall Problem.

Monty Hall's "Let's Make A Deal" is gameshow where you choose a door to win either a goat or a car--the objective being to win the car.

Wikipedia has a pretty thorough article on the Monty Hall Problem, but I want to make my own little problem and proof using PMI, because I wuvvy wuv wuvs PMI so much, yes I do!


Cars vs. Goats (AKA: Monty Hall Problem)

Suppose there is a gameshow in which the player chooses a door. Behind any door is either a goat, or a car; and the objective is to pick the door with a car. After a door is chosen (but not opened), the host opens a door, behind which is a goat. Theplayer must then make a choice: stay with the chosen door, or switch to another door.

More specifically, let's deal with the case where there are c doors and only 1 car. That means, there are c - 1 goats, and the player can choose to switch to one of c - 2 remaining doors, with the objective of finding the car.

Scenario One: c = 3
In this case, there are three doors, and suppose the car is behind door three. There are three initial choices the player can make: door one, door two or door three. Then, the player can choose to either switch or stay.

Here are the results:

initial:	1	2	3
switch:	Car!	Car!	Goat
stay:	Goat	Goat	Car!

If we look at the switch row, we see that 2/3 result in cars.
If we look at the stay row, we see that 1/3 results in a car.

Thus, it is better to switch when there are three doors and one car.

Scenario One: c = 4
Let's suppose the car is behind door number four. The only real difference now is that there are two possible switch choices, but that's easy to take into consideration.

Results:

initial:	1	2	3	4
switch1:	Car!	Car!	Car!	Goat
switch2:	Goat	Goat	Goat	Goat
stay:	Goat	Goat	Goat	Car!

Looking at both switch rows, we see that 3/8 result in cars.
Looking at the stay row, we see that 1/4 results in a car.
Since 3/8 > 1/4, switching is still a better choice.

The remaining cases
Let's skip over the details of the next few cases. Let S be our probability of success in finding the car when choosing to switch, and c be the number of doors; still having only one car.

Summary:

c	3	4	5	6	7
S	2/3	3/8	4/15	5/24	6/35

Noticeable patterns:

• S_c = (c - 1) / [c (c - 2)]
  To simplify things let's separate the Numerator and the Denominator,
  
  let N_c = c - 1
  
  let D_c = c (c - 2)
• N_(c+1) = N_c + 1
  
• D_(c+1) = D_c + 2c + 1

Proof of pattern of S:

1. Prove that N_(c+1) = (c + 1) - 1

First, look at the best case of N_c = c - 1, ie: when c = 3

N₍₃₎ = (3) - 1

N₍₃₎ = 2

This is true!

Now prove all following cases are true; ie: prove N_(c+1) = (c+1) - 1

N(c+1) = Nc - 1
N(c+1) = (c - 1) + 1	Substitute.
N(c+1) = c	Simplify. Hoorays!

2. Prove that D_(c+1) = (c+1)(c+1 - 2)

First, look at the best case of D_c = c (c - 2), ie: when c = 3

D₍₃₎ = 3 (3 - 2)

D₍₃₎ = 3 (1)

D₍₃₎ = 3

This is true!

Now prove all following cases are true; ie: D_(c+1) = (c+1) (c+1 - 2)

D(c+1) = Dc + 2c - 1
D(c+1) = [c (c - 2)] + 2c -1	Substitute.
D(c+1) = c2 -2c + 2c - 1	Expand.
D(c+1) = c2 - 1	Simplify.
D(c+1) = (c + 1) (c - 1)	Difference of Squares. Hoorays!

Therefore,

S_c = (c - 1) / [c (c - 2)]

This doesn't really mean anything relevant yet, though. We've just proven that we always know our success rate, and not that switching is better than staying. Let's look at the staying data now.

Let T be the success rate of staying, and we'll compare with S.

c	3	4	5	6	7
T	1/3	1/4	1/5	1/6	1/7
S	2/3	3/8	4/15	5/24	6/35

Pretty dismal, eh? Let's prove that S > T. But first, we must prove that T always follows a pattern...

Patterns noticed in T:

T_c = 1 / c

T_(c+1) = (T_c) * c / (c + 1)

Prove that T_(c+1) = 1 / (c + 1)

Best case: c = 3

T₃ = 1 / 3

This is true!

All following:

T(c+1) = (Tc) * c / (c + 1)
T(c+1) = [1/c] * c / (c+1)	Substitute.
T(c+1) = c / [c (c + 1)]	Multiply through.
T(c+1) = 1 / (c + 1)	Simplify. Hoorays!

Prove that it is always better to switch:

Here's the final piece! We must prove that S_c > T_cProof by Contradiction.
Go!

Sc	<	Tc	Assume.
(c - 1) / [c (c - 2)]	<	1 / c	Substitute.
c (c - 1) / [c (c - 2)]	<	c / c	Multiply by c.
(c - 1) / (c - 2)	<	1	Since c > 2, this is most definitely false! In fact, (c-1)/(c-2) cannot even be equal to 1.  It is always greater than 1.   Since all our steps after the initial assumption of Sc <>c, that assumption must have been incorrect. Therefore, Sc > Tc  for all integers c > 2. Hoorays!

Thus, it is always better to switch.


I'll write something more journal-like later.
--Charissa

Dreams

Around spring this year (2007), I noticed I wasn't feeling the same. I felt normal, but, somehow, less something, and consistently so. So I went to the doctor, and was prescribed thyroid hormone.

One of the strangest "side effects" happened to me: lucid dreaming.

I'd wake up in the morning, not knowing what to think anymore. When my dream-experiences contradict my waking paradigm--which they often do--then I'm left totally confused, because my dream-experiences feel so valid!

Most notably, I've had ponderous remarkable dreams about James.


Backstory

James and I met in Honours Calculus last September, and we got along quite easily. We started dating in December, and it was totally awesome geek-love, complete with writing geeky love notes to eachother. This was actually during my coming-out as "curly", for the first time, to close friends. James didn't mind, though, because he had a close friend, Kay, who identified as a lesbian.

Spring rolled around, and the school year was ending. I had noticed that James was beginning to withdraw, and perhaps actively hold something back from me; but his silences had always confused me, and I wondered if it was just nothing. On the last day of school before exams, we talked, and he told me he had developed feelings for another girl. We decided to take a break; especially with exams coming up and the school year ending; and we'd talk more about this after some time apart.

About a week later, we decided we'd be friends from now on.

We kept in touch a little, mostly through e-mail, though we met on occasion over the summer. He and Kay started going out. It was a little amusing for me to wonder--because she had thought I was attractive, and James and I were dating, and I'm bisexual--about all sorts of possibilities.

And over the summer, James was planning on moving into the city with Kay.


The First Dream

The first dream I had was terribly confusing; mostly because I'm not sure if it was me leading the dream, or otherwise.

I'm not sure by what circumstance, but James and I were talking in the dream. I think we were doing something romantic. Suddenly, I stopped, and I asked something about Kay.

James does this thing with his posture sometimes, like closing up; and I've come to interpret it as a sign he's sad, distressed, scared. He did that, in the dream, and I asked him what was going on. He explained something about Kay, though I don't remember what, and I got the sense he was worried, but also hurt or offended or outraged and something like he wanted to do something about it but couldn't, or didn't know what.

A few months later, I visited James at his new address in the city. After some general chatting, I asked where Kay was; and he explained that her parents were not permitting her to either move out of her house, or into this one. And he did the same thing he did in my dream--the closing up--and I got the same readings from him.


The Second Dream

After hearing about Kay, I was even more rattled about the dream, because it had started off with a very close and probably romantic feeling between me and James. I don't know what to make of this, but the second dream was almost entirely romantic.

In this dream, James and I were swimming, not just in a beach, but something that seemed farther, larger and deeper. It reminded me of Venice: salty ocean water.

I don't even remember all the details, and there are some I don't want to mention; but we kissed, and there was a time he asked if this was okay--because of this, that and the other thing--and I said yes, so we kissed again.

Well, I guess it wasn't okay, because I woke myself up immediately; and I was frightened and bewildered.


Charissa probably thinks too much

I don't know what to make of these dreams.

Probably the most likely explanation is that, if these dreams mean something true about James, they mean that James still has affections for me (as I do for him), but we simply cannot do anything about them because of the choices we've made and the lives and responsibilities we have now.

Or, I'm simply projecting what I want, and just happened to be right about Kay being in some sort of trouble that affected James in a very predictable way.

I wonder how much of the first dream was meaningful, though, and if the relevance of one dream is an indication of the relevance of all other dreams--ie, if one dream came true, maybe all my dreams can come true.


With my birthday party coming up, I'm wondering whether or not to invite James. I mean, I want to, because I still want to be friends with him, but he still affects me too deeply, maybe--too deeply to know if it's real.

I'm going to try meditating tonight.
--Charissa

Dressup!

I was at work, wondering what I'd like to dress up as for Halloween. Even though I'm too old for trick-or-treat-ing, I still like to dress up for fun.

Last year, I went to University as a drunken wench--bottle and everything--which was neat, because we got our assignments back that day, and I got 100%.

I couldn't decide on a costume.

So I pondered...	...and pondered.

Aha!

I decided I'd dress up as my dad!

Actually, that's not entirely accurate. I decided I'd dress up as one of those old-days Chinese martial artists -type guys; but when I grew out my goatee, I looked disturbingly like my father.

How appropriate for Halloween!

--Charissa!

Charissa thinks too much

Kevin came over tonight.

Kevin went to school with my good friend Mark, and I first met Kevin at Mark's sixteenth birthday party. We didn't see eachother again until this year; so a lot has changed since then. Mark and Kevin came over to watch a movie tonight.

Right now, Kevin's majoring in English, and has been interested in film for a while--so he's a Theatre Kid! He's also very much a Virgo, and a fun person to be around.

Do I like him?

Yeah, definitely; I mean, he's a great guy and all.

Do I like him?

That's a bit more complicated. I like spending time with him, and I love that he knows how to move/mime/act; he's graceful and articulate that way. And, sure, I think he's physically attractive. The truth is that, yes, I like him; but for one trait. He's such a Virgo. I mean that he's not very passionate; he's a perfectionist for the sake of being a perfectionist, not because he is compelled.

On a side note, I guess this also says something about me. I like to be able to affect people--especially romantic partners. I'm not sure if this means I have some kind of power complex where I need to be able to control people; but in any case, I like to know I cause change or reactions. (This actually caused me some problems in my last romantic relationship, because my sweetheart wasn't a very expressive person, and so I had difficulty interpreting the lack of responses.)

Anyway.

Coupled with the fact that I have never been prepared for romantic relationships, is the knowledge that Kevin is so completely not appropriate for me.

I wish I had the ability to let myself simply enjoy whatever -ship we have or will have; but I'm so worried I'll do something stupid--again. I wish I could run my fingers through his incredibly wiry hair and tell him how handsome he is. I wish I could tell him how much I admire his ability to express with motion. I wish I could kiss him. And then, I wonder why I want all these things.

The words "boyfriend" or "girlfriend" never meant anything real to me; mostly because every time I was introduced to someone's -friend, they spent the rest of the time doing private things together in our company, so maybe I've associated those words with those private actions--kissing, playing with hair, affectionate petting. So, somewhere in my mind is this picture of what a romance is supposed to look like. I think the whole problem is just about labeling; but I can't simply not-label anything. That's the opposite of learning.


My best friend 
 
This is how Eric and I became best friends.

We met eachtoher in the winter of 2005 when our high school's choir needed more males. Later, our Music groups took a trip to Chicago on a bus. Eric and I were bus-buddies, which basically meant that we sat together on the bus for the 17 hours there, the 17 hours back, the umpteen hours traveling from one destination to another, and spent umpteen hours together at the actual destinations. The only times we weren't together was when we slept in different rooms (boys with boys, girls with girls; though this arrangement presumes a bit) or when one band went here while another went there.

So, on that trip, we became pretty comfortable with eachother.

However, it was the Summer of Eric that made the difference.

Over the spring and summer, we spent time together; mostly biking. When the summer of 2006 rolled around, and I'd graduated, we still managed to find time. We spent many summer nights just sitting, standing or laying on his front lawn and talking; sometimes as last as midnight (which was late by his standards).

During this time, we talked about anything: God, ourselves, family, relationships, people, bands, music, our futures... I think Eric was the first person I "came out to" (I remember saying, "What would change if I started calling myself 'bisexual'?"). Or maybe Mark was first, I forget.

In any case, we developed an ease of understanding eachother; of communicating and reading.

I used to make this gesture whenever I found something unbelievable: I'd put my hands just behind my head, with my fingers touching my thumbs, and then bring them forward suddenly and open my palms. The first time I texted "?!" to Eric, he replied with a lol, saying that he could see exactly what that expression looked like.

One night, Eric and I were talking, and one of us wanted to ask a question, and the other said, "Yeah," right away, and we both understood exactly what the other wanted to convey. And then Eric sort-of paused and said suddenly, "I think this is what marriage must be like."

We talked once about "going out", and we both have our own reasons--whether or not they coincide is of little consequence--but my strongest reason is that, to have any other relationship with him, now, after our best-friend-ship, would be less. It just--it wouldn't.

Eric currently has a girlfriend, and I was extremely happy when I found out (he'd never had a girlfriend before, and felt he was missing out; maybe because I had a sweetheart at the time and prattled on and on, and on, and on). Eric explained to me that, right off the bat, they had talked about what their relationship would be like; and they decided that it didn't matter what "girlfriend" or "boyfriend" meant; they were going to do as they liked, regardless of others' expectations.



This is what I would love to have with Kevin--no expectation. If we're friends, we're friends; if we laugh, we laugh; and if we kiss, we kiss.

What disturbed me about my affections for him was that he is so completely not what I'd prefer in a lover. (Ideally, I'd want a brooding math genius; but that's for an entirely different blog entry.)

So, I think I need to learn that it's okay to be however I am; it's okay for him to be however he is. ... I hope I learn that quickly and well!



Other things I've thought about tonight:

- This was the first time Kevin was in the house. I showed him around, and he commented, twice, "This is a big house." So after he left, I walked around and stood where he had stood and tried to see as he might have seen. (Both he and Mark come from much smaller houses than this.)

- I was talking with Mum after they'd left; and suddenly said, "Man, I don't think I could have kids. I mean, I don't understand the kids one year younger than me; and we--you're thirty years older than me, and we can barely communicate! How could I ever hope to communicate with people thirty years younger than me if I can't understand the kids one year younger?"


And that's all for tonight. As the title of this entry so plainly states, Charissa thinks too much.

--Charissa