From my personal statement, part of my application to grad school:
Midway through my first probability course I began seeing the world in a different way. Absolutes gave way to probabilities, certainty to estimation, and placid acceptance to examination. Phrases like “mutually exclusive” crept into my daily parlance and the salad bar became much more interesting when order mattered.
To illustrate an interesting principle, let me summarize a problem from the Statistics: Concepts and Controversies lab book. I can't remember the actual numbers, so I'll make some up. The problem is a critical look at an article put out by the Associated Press. The article is basking in the improbability of a certain event. The event is that the past 16 births at the local hospital in some small town in Indiana have all been boys. Well, surely, if the probability is about 50% for either gender, we should have seen about 8 boys and 8 girls, right? The probability is not difficult to calculate. Assuming that the sexes of the babies were independent, the probability of 16 boys in a row is .5 to the 16th power. 0.5^16 = 0.00001525. If you were a gambler you wouldn't bet on it, right?
So the question is this: Is this story newsworthy?
Counter intuitively, the story is not newsworthy (in a statistical sense).
Why isn't the story newsworthy? Four years ago, according to the 2005 census, there were 7,569 hospitals in the USA. In every hospital across the country, children are being born. Is it so unlikely that one of those hospitals gets a run of male births? Think about it another way. Let's say you go to a baseball stadium with 7,568 of your closest friends. You all crowd onto the field and each of you pulls out a coin. I call out "go" over the loudspeaker and everyone starts to flip his or her coin. Each of the 7,569 people flips a coin 16 times and records the number of heads and tails.
After everyone's finished, someone in the middle of center field yells out "Holy Balls, I just got 16 heads in a row!"
Is this so unusual? Do you grab this guy, throw him in your trunk, and make him play craps for you in Vegas? For legal (and statistical reasons) I wouldn't. The punch-line is that if you work on a large enough scale, even outcomes with very small probability will eventually occur. Now, if you do the math, the probability that at least one person gets 16 heads in a row (or at least one hospital gets 16 males births in a row) is about 89%. That means that if I wanted to, I could find a hospital almost every day that had this exact same, "newsworthy," event.
About 10 years ago, a man won the lottery...for the 2nd time. Holy shit, you may exclaim, certainly Baby Jesus has blessed this man above all others. Ye, let us make a shrine to him and rub its bronze nose for luck before we watch the powerball drawing. Again, faulty logic. The fact is that many people play the lottery, and, in fact, many people who win the lottery keep playing afterwards. If you get enough repeat players, eventually one of them is going to win again. Some statistician claimed it would be about every 10 years, and about 8 years later someone else became a repeat lotto winner.
Now I'm getting long winded, but I want to give one more example of how this logic applies. Say that you're talking to a new coworker and after a while you find out that you both had golden retrievers as kids and that BOTH of you have sisters named Julia. Is this crazy? Are you soulmates? People ask me, what's the probability that you meet someone at random and you both have sisters with the same first name? It's not quite the correct question. You see, if you talk for long enough, you're bound to find something in common. So when you do, you tend to forget about the dozens of things you don't have in common, and fixate on what you share. That's fine, since that's how we develop interpersonal relationships, but it's not nearly as "weird" or "spooky" as it might appear to be.
blah blah blah, i'm done. did anyone find this interesting?
3 comments:
Yes, I did. You are extraordinary, SprocketPlug.
Sq
yes.
This reminds me of a conversation I had with someone about about flipping a coin 100 times. If it came up heads 99 times, the probability of it being heads on the 100th throw is still %50. While throwing heads 100 times in a row is pretty unlikely, throwing heads just one more time is always %50. Anyways at the end of the conversation the person still did not believe me, because the concept is a bit counter-intuitive I guess.
Rereading that, this actually had nothing directly to do with your post except that you took a concept that is counter-intuitive and did a good job explaining it!
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