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CCC comments on Welcome to Less Wrong! (July 2012) - Less Wrong
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Fortunately, it's also very easy to get a basic grip on it. Multiplication, addition, and a few simple formulae can lead to some very interesting results.
A probability is always written as a number between 0 and 1, where 1 is absolute certainty and 0 cannot happen in any circumstances at all, no matter how unlikely. A one in five chace is equal to a probablity of 1/5, or 0.2. The probability that event E, with probability P, is false is 1-P. The chances of independent events E and F, with probabilities P and Q, occurring in succession is P*Q. (This leads to an interesting result if you try to work out the odds of at least two people in a crowd sharing a birthday)
Probability theory also involves a certain amount of counting. For example; what are the chances of rolling a seven with two ordinary, six-sided dice? (Assuming that the dice are fair, and not weighted).
Each dice has a one-in-six chance of showing any particular number. For a given pair of numbers, that's 1/6*1/6=1/36. And, indeed, if you list the results you'll find that there are 36 pairs of numbers that could turn up: (1, 1), (1, 2), (2, 1), (1, 3)... and so on. But there's more than one pair of numbers that adds up to 7; (2, 5) and (1, 6), for example.
So what are the odds of rolling a 7 with a pair of dice?
Yeah, it's the counting problems that I've been avoiding! Because there are some that seem like you've done them correctly and someone else does it differently and gets a different answer and they still can't point out what you did wrong so you never quite learn what not to do. And then conditional probabilities turn into a huge mess because you forget what's given and what isn't and how to use it togetherrrr.
I hope it's a sixth, but at least this question is small enough to write out all the combinations if you really have to. It's the straight flushes and things that are murder.
Ah, I see. You'll be glad to know that there are often ways to shortcut the counting process. The specifics often depend on the problem at hand, but there are a few general principles that can be applied; if you give an example, I'll have a try at solving it.
It is, indeed.
In fact, many if not most concepts in probability theory deal with various ways of avoiding the counting process. It gets way too expensive when you start handling billions of combinations, and downright impossible when you deal with continuous values.
asdfjkl; I wrote out all the pairs. -_- Can't trust these problems otherwise! Grumble.
"You are never too cool to draw a picture" -- or make a list or a chart. This particular problem is well served by a six-by-six grid.
Dice are okay; it's the problems with cards that get toooo huge. :)
Can you give an example?
I will try to hunt one down! It's usually the problems where you have to choose a lot of independent attributes but also be careful not to double-count.
Also, when someone explains it, it's clear to see why their way is right (or sounds right), but it's not clear why your way is wrong.
Yes, I notice that people are in general either bad at giving or reluctant to give this kind of feedback. I think I'm okay at this, so I'd be happy to do this by PM for a few problems if you think that would help.