****** OT ****** (Sorry folks, I was a little late with this post and didn't see that it had already been beaten to death in the prior 50 or so posts.)
5Speed, you're the first catch of the day!
Bayes' Principle is a mathematical formula that addresses this problem. In plain English, it says that the probability of a particular event occurring depends on how many parts of that event have already occurred.
To illustrate, the chance of rolling a 12 with a pair of dice is 1/36 (there are 36 different combinations of two sets of 1 through 6, and only one of those (6 and 6) adds up to 12. By contrast, the chances of rolling a seven are 1/6, much higher, because there are 6 ways to roll a seven (1,6; 2,5; 3,4; 4,3; 5,2; 6,1) -- they didn't pick seven just for kicks when they designed craps).
The chance of rolling a 12 four times in a row is (1/36)^4, or 1 in 1,679,616 (terrible odds). You and I agree about that.
BUT the chance of rolling a 12 four times in a row, given that you've just rolled 12 three times and you're about to roll the fourth time, is 1/36 (the same as rolling a 12 once). That's what many people don't understand, and that's what Bayes' Principle tells you. You can adjust the probability of an event occurring based on what's already known about the component events.
Think of the four rolls as four sub-events. The probability of each one occurring is 1/36. If none has occurred yet, the probability of the entire event -- four 12's in a row -- is (1/36) x (1/36) x (1/36) x (1/36), or (1/1,679,616). But if you roll a 12 on the first roll, you've satisfied the first sub-event -- we know it occurred, so the chances of it occurring are 1 in 1. So now the chances of four 12's in a row are 1 x (1/36) x (1/36) x (1/36), or 1 in 46,656. If you roll a 12 on the second roll, it's now 1 x 1 x (1/36) x (1/36), or 1 in 1,296. Roll a 12 on the third roll and it's now 1 x 1 x 1 x (1/36), or 1 in 36. Roll it on the fourth and it's 1 x 1 x 1 x 1, or 1 in 1, or certain to occur because it did. There's no magic going on here, it's just recognizing that the likelihood of finishing a task grows more likely as you complete more and more of the task.
Example the Second: Today a March 1998 $100 call on ASND is probably about one cent. But if the stock rose to $98 tomorrow, the call would be quite a bit more valuable. That's a really flawed example of the principle, but it helps illustrate the point. You can't get to $100 unless you first get to $50, $60, $70, and so on. $100 gets more likely as you pass the obstacles.
Example the Third: Suppose an HIV test has a false-positive error rate of five percent, meaning that, on average, five percent of its positive results will be wrong (i.e., tested person doesn't have HIV). Suppose further that in the country of Transylvania a person has a 1 in 100,000 chance of having HIV. Now suppose there's a 90-year old chaste Transylvanian nun who's never had surgery, never been given an injection of any kind, and has never engaged in any so-called "risky" conduct. She takes the test and it comes out positive. Does that mean she now has a 95% chance of having HIV? NO! NO! NO! NO! The chance is still virtually nil, and the chance that this particular test gave a false positive is 99.99999%, not 5%.
Why? The five percent error rate is an independent event, like a roll of the dice. It says nothing about the person taking the test -- the test doesn't know the person's history. To be correctly diagnosed by this test as having HIV, the nun would have to (1) become infected with HIV (1 in 100,000), and (2) get a true positive result on the test (95 in 100). In that case, Bayes' Principle takes the original event (correctly diagnosed by this test as having HIV), looks at the probabilities (1/100,000) x (95/100), and replaces it with (1/1,000,000,000,000,000,000,000,000) x (95/100), because we're virtually certain that sub-event #1 did not occur (we can't say it didn't, because maybe she stepped on an infected needle during charity work back in 1983, which I'll give odds of 1 in 1,000,000,000,000,000,000,000,000 of happening). So REGARDLESS of the test result, we are pretty sure she doesn't have HIV. (Math-heads out there, I know that the 95 in 100 statement isn't right, but it's close enough to illustrate my point.)
Still not convinced? Ok, answer me these questions:
1. Do the dice remember what they've just rolled? Does one say to the other "Enough with the sixes, already!?"
2. What if we roll three 12's, go walk around the block, and then roll the dice again? Have the dice reset themselves during the delay? How did they know? Does it make any difference if you cover up the dice so they can't see you, or if you leave quietly?
3. Suppose at the die factory they rolled the dice 300,000 times and each time they came up seven, and then they packaged them and shipped them to the store. Then when you buy the dice and roll them, is seven less likely than 1 in 6 to come up?
4. Why does Las Vegas close sports bets after the game starts? If it's always the case that four 12's in a row have a 1 in 1,679,616 chance of happening, even if you've already rolled three of the four 12's, then why can't I just wait until right before the last roll before placing my bet? And why can't I wait to see what the Super Bowl score is probably going to be at the two-minute warning before placing my bet? Same thing.
By the way, 5Speed, I've read the later posts since your first and realize that you probably already understand what I'm saying. But no way am I throwing out this chef d'oeuvre of a post.
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I think you're wrong, Sowbug.
The probability to get '12' four times in a row is 0.00005 ((1/12)^4). The probability to come up with something else is 0.99995. Those guys who didn't believe were right. It's not very probable that you throw a '12' four times in a row. |
