Tuesday, March 23, 2010

By the Numbers

I presume most people recognize that there is a vague connection between statistics and probability, but, having taken a course in probability theory, I'd be willing to bet the farm that very few people realize the full breadth of intimacy between the two. This is true in particular because despite having studied both, I'd count myself as one amongst the naive. From the outset probability is simply difficult, and often counter-intuitive. Not only does probability proceed in ways contrary to our intuition, it does so in such an amazingly tricky way! Maybe it is a function of how easy it starts out: given a typical six sided die, most everyone knows that the chance of guessing which number comes up is one in six. Easy enough, you pick one side out of a total 6, so the probability is 1/6. The common understanding of probability stops there, for the simple reason that any situation even marginally more complicated than that becomes remarkably more logically and mathematically sophisticated. Suppose I'm flipping a coin and you're guessing the results. For some reason you're having terrible luck and you've guessed wrong 10 times in a row, what's the probability that you guess the next flip wrong as well? Think about it for a minute and when you've logically arrived at what must certainly be the answer, highlight the following space for the answer:  1/2

Next, try to logically deduce the probability of guessing incorrectly for 10 coin flips in a row. Answer:   1/1024

It only gets so much worse from there, to the extent that I'm really not confident I could present the correct answers myself! Even admitting that I can't help but try for one more. Assume that 4 out of 5 people prefer Crelm toothpaste. What's the probability that from a selection of 5 people 4 of them prefer Crelm? Answer (I think): 256/625

The important notion here is that a probability says something both nebulous and concrete about reality. If a truly random die is thrown 6 million times, in all likelihood each number will have come up about 1 million times. If 4 out of 5 people really do prefer Crelm, then the chance that a randomly selected person prefers Crelm is 4/5 or 80%. As much as we all like to think that the statistics don't apply to us (because we're special), if the statistics are accurate there's no way to escape them. Most of the time this is a banal statement, as when referring to whether or not you prefer Crelm--either way it's not exactly a big deal. But then... there are the other statistics. "Around 50% of US marriages end in divorce" can be a pretty hard pill to swallow for a couple walking down the aisle. I have reason to believe the number of couples who'd figure they end up on the successful half of that statistic while exchanging vows is much higher than 50%--clearly if they thought it wasn't going to last they'd probably not be entering the commitment in the first place. Similarly, doubting the success of the marriage from the outset probably isn't going to increase the chance of a favorable outcome. What's left is an awkward position, objectively maybe the best one can think is that at least the odds aren't as bad as they could be, better than any casino game. However marriage is a particularly special case for a number of reasons, the primary one being the shift in locus of control which is applicable to all interpersonal relationships; though a bit less severe, anyone who's been dismayed by the lack of a second date (etc.) knows the score. To be fair the actual divorce rate changes based on many factors, where 50% is just the overall rate. The lowest divorce rates are found in each of the following categories: first marriage, atheist or agnostic, age 30 or older, residing in the Northeast and no cohabitation prior to marriage.

Uncontrollable statistics naturally lead to other more personally manageable probabilities. For instance, 28% of car accidents in the US happen while at least one of the drivers is using a cell phone. This is the part where I reiterate: we love to think we're special and that the statistics don't apply to us, but it just doesn't work that way. We are all special, I'm fully on board with that, but that doesn't grant any of us statistical immunity. Using a cell phone while driving (even with a hands-free headset) substantially increases the chance that you will be in a car accident, which could result in your death, or, arguably worse, the death of another/others with the accrual of manslaughter charges and the lifelong burden of knowing that you've killed someone. It's very simple: while the car is in gear, your phone doesn't exist. There are absolutely no excuses.

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