Innumerate?
Posted by seed @ 7:19 AM
Quick side note: My wife and I took an online GMAT pre-test, back in 2001, I think. Things are spicy around the seed household on a Friday evening. It took us three hours and I found myself trying to rationalize and solve equations like that from scratch, without remembering how to actually remove a variable from an exponential factor. I tried this and that with numbers I could handle in my head. Soon, immense pain set-in and I had a new appreciation for Pathagorus. Anyway...
An interesting concept came out of the book that reminded me of a conversation I had with either Bergeron, or Savage. I cannot recall which. But first, a warm-up...
Say you walk by a roulette table that has a display that reads seven consecutive reds. What's the chance that the eighth spin will result in a red? Quick math for eight consecutive reds results in this: (18/36) x (18/36) x (18/36) x (18/38) x (18/38) x (18/38) x (18/38) x (18/38), or (18/38)^8. 18 being the number of reds out of 38 roulette slots. 8 being the number of attempts. Simple enough, right?
I get .00025 or .025%.
So, you might be considering putting your money down on black at this point. Clearly, twenty-five thousandths of a percent are pretty damn good odds; fabulous by most standards. Obviously, the roulette wheel is not a conditional probability; every spin is independent of the last. So, when you walk up to the display and see the seven previous reds you may forget that and think about the probability of the eight red. You also forget that you walked up to the table after the seventh spin, and they've all been red. So, you are already here: (18/36) x (18/36) x (18/36) x (18/38) x (18/38) x (18/38) x (18/38). What's the probability of the eighth spin being red, at this point? You bet: 18/38. Almost even money.
Warmed up yet? Good. I recall a conversation that went something like this:
Them: A certain medical test is 98% accurate.
Me: Yeah...?
Them: But it can give false positives that are close to 20%.
Me: Whaaaaaaa? No way.
I may have been innumerate at that point in my life and not have realized it. Which might be ignummerant. Either way, here's the skinny:
The test is 98% accurate, as stated. So out of 10,000 trails it results in 9,800 positives and 200 negatives. What happens if a condition is added to the mix, like our roulette example above? Then, the condition was that we walked up to table with seven previous reds, that greatly increased the probability of an eighth. Add to the test the condition that .5%, or 1 out of 200 cases, are confirmed positives among a population. How does that change your probability of actually being a positive, if you received a positive result?
Try this: .005 (.5%) x 10,000 = 50. This is the number of real positives in a sampling of 10,000. But we're looking for false-positives, so those 50 are removed from the 10,000 that are given the 98% accurate test, yielding 9,950. How many of those will be false? 2%, right? Which is 199 (9,950 x .002). Add that number to the known condition: 50 out of 10,000 are positive. You get 249. Your chance of actually being a positive out of all the positive results is 50/249, or about 20%.
I know, I was lost too.
Comments
98 percent of life is half mental.
(WIth apologies to Yogi.)
Posted by: ~Easy | March 14, 2007 12:33 PM
Funny. I'm reading it right now as well. It's a pretty interesting book. I tend to think of myself as decently versed in mathematics, but the scope of the really big numbers is sometimes hard to grasp without example.
"For example, knowing that it takes only about eleven and a half days for a million seconds to tick away, whereas almost thirty-two years are required for a billion seconds to pass gives one a better grasp of the relative magnitudes of these two common numbers. What about trillions? Modern Homo sapiens is probably less than 10 trillion seconds old;"
We get daily doses of these numbers. A thirty million dollar jackpot. A billionaire ass-bag with his own tv show. A multi-trillion dollar federal debt....
Every educated person knows they all differ by a factor of one thousand, but the great thing about the book is it's ability to break down the commonality of the numbers as their own distinct entities and represent them in such a way that forces you to examine the true scope of them.
I'm only a few chapters in, but so far, so good.
Posted by: dutch | March 16, 2007 5:01 AM
For anyone that wasn't scared off by seed's post, there's another way to think about the test thing. (Actually, it's pretty much the same, but gives the whole thing a structure.)
It's called Bayes' Theorem. The Wikipedia thing is long, but interesting. It essentially gives you the same sort of result as seed has here.
What most people don't consider when talking about tests being X% positive, is precisely that false-positive/false-negative issue. And it comes up more than you would think. One extreme example is my own experience with a lie-detector. The people administering the polygraph tout that the thing is something like "85-90% accurate". But bring up Bayes' Theorem, and they stare at you like you just admitted to killing someone. With a test at 98% accuracy with a 2% false-positive rate, your (random) chance of being positive is 20%. Imagine what that gets to be when the false-positive rate is 10 times that: i.e. 20% false positive/negative if we take that polygraphs are 80% accurate. The chance that you've lied is pretty much a coin toss.
No wonder we didn't catch Hanssen at the FBI through those things.
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