The grass really is greener!

Tyeliah recently finished her first class - probability - or rather the first half of the probability and statistics class (there are 2 different teachers for the two halves, so I call it two classes). To celebrate their last lecture with the probability teacher - Arnie - they all wore orange - Arnie's favorite color. They also all created new name placards.

I just went to grad school. I didn't have name tags. Hmmm...

Anyways, instead of being 'Tye' - Ms. Duncan's business-y name - she went by 'Tray Table Catcher.' I have been informed that Ms. Duncan would have been much more creative had she known that I would be blogging about the experience. Apparently The Tray Tables were a baseball team in a homework problem. And I thought the Banana Slugs was the worst mascot name.

Tyeliah thoroughly enjoyed her time in probability class - the teacher was quite entertaining or so I'm told. I've always been a fan of probability myself, being somewhat mathematically inclined. I also had an excellent probability teacher in my undergraduate days, a swiss man who had an uncanny resemblance to Santa Claus who would come in every class and ask "Does anyone have any questions today" and when inevitably no one would say anything (college is supposed to be a place of flourishing debate but I think anyone can attest to the fact that there are an inordinate amount of uncomfortable silences) he would say "Good - my dog ate all the answers so I wouldn't be much help" or some other ever varying excuse about how he had misplaced the answers. I enjoyed it.

Anyways, a curious probability quandary involving expected values was recently explained by the probability and statistics TA on the way back from a recent plant trek (more on that later). An expected value can be explained as follows - say you get $60 if you roll a 6 and $30 if you roll a 5 and $0 if you roll anything else - you have a 1/6 chance of rolling a 6 and a 1/6 chance of rolling a 5 so the expected value of one roll of the dice is (1/6*60+1/6*30+2/3*0) $15. If you think about it in casino terms, the casino would want to set up a game where you had to bet greater than $15 to roll the dice so they would be assured (in the long run) of making money. I'm sure there's a better way of explaining expected values, but you get the idea.

The TA created a problem as follows. Say you are presented with 2 envelopes - one of the envelopes has twice as much cash as the other - although you don't know the value of the cash in either envelope. You choose envelope A and open it up to find $100. So there's a 50% chance that you chose the higher valued envelope (meaning that envelope B has $50) and there's a 50% chance that you chose the lower valued envelope (meaning that envelope B has $200). Then you're given the option of keeping envelope A or switching to envelope B. What do you do?

The choice rests upon the expected value of envelope B - if the expected value is greater than $100, you should switch. So what's the expected value? 1/2*50+1/2*200=$125. Therefore, you should always switch to the other envelope.

Thus the grass is always greener on the other side of the fence and if given the option, you should always jump over the fence.*

Personally, I love the liberal arts and sciences for many reasons but mostly because I like the idea that any triviality can be analyzed and analyzed and analyzed again and connected to deeper truths about our inner psyches or broad all encompassing truths about the nature of our societies. So, in my love of overanalyis, I decided to research the question of whether the grass really is greener on the other side.

Lo and behold (a phrase worthy of its own analysis) I came across this, an academic paper purporting to show that the grass may not actually be greener on the other side but that we are psychologically disposed to think so. It also explains why I always seem to choose the slowest lane in the grocery store or make all the wrong choices as I attempt to navigate a traffic jam.

Ahhh, internet. You never cease to amaze me.

*There are of course problems with this probability paradox - something to do with the lack of an actual upper limit value on the amount of the cash in the envelope which if it were to exist (as it empirically must) would take away from the higher expected value of always switching. But fun nonetheless, wouldn't you agree.**

**Ms. Duncan believes I just made up this problem with the probability paradox and that it is not actually true ... she may be correct ... it's been known to happen in the past