Map of life expectancy at birth from Global Education Project.

Wednesday, April 21, 2010

At least something is normal around here

Sometimes people do this bit using a magic bag containing an infinite number of red and green marbles, but I'm just going to flip some coins. We'll let a magic guy do it to make it interesting anyway.

Heads are H, and tails are T. That's easy. And we know that if you flip a coin once, the probability is 50% heads, and 50% tails, so P(H)=.5 and P(T)=.5

So what if you flip a coin twice? What's the probability of getting two heads? (Jeopardy! music plays for 15 seconds.)

Okay, as you remember from last time, the two coin flips are independent events -- the first one has no influence over the second one. (Believing otherwise is called the "gambler's fallacy" -- you start to think your number is "due" because it hasn't hit for a while.) So you get the probability of two independent events both occurring by multiplying the individual probabilities together. The probability of heads the first time is .5, and the probability of heads the second time is .5:

P(H H)= .5 * .5 = .25

It will happen 1/4, 25%., .25 of the time. Two tails in a row will also happen 1/4 of the time. As you may also remember, the probability of either one of two mutually exclusive events happening is the sum of the probability of each.

P (H H) + P (T T) = .25 + .25 = .5

As we also know, something has to happen, which means that all the probabilities have to add up to one, which means that what is left -- 1 heads and 1 tails -- must have a probability of .5. That makes it, obviously, twice as likely as TT or HH. Why is that? Because there are two ways it can happen: tails then heads; and heads then tails.

P (T H) = .25 and P (H T) = .25, so the probability of either one happening is .25 + .25 = .5. I'll even show you a graphical representation:

Okay, let's keep going. Let's do four coin flips.

You should be able to figure out for yourself how to compute the probability of each possible combination. There is a general formula for this, which is called the Binomial Distribution -- binomial meaning "two names," or "two things." I will show it to you just for the halibut, but it isn't necessary and you don't need to know it. (I can't show it in this post because I haven't been able to get the terms in the formula to line up correctly. Blogger is not very math friendly.) What you do need to know is the limit of the binomial distribution, i.e., what it starts to look like as the number of coin flips gets very large. Here it is:

I put some real data under the curve just to show you that in the real world, it never comes out exactly right -- that's the way it is with probability. But do you recognize that curve? Yes, it's the world famous Normal Curve, the Bell Curve. It is the basis for an entire approach to statistics. It was discovered by a guy named Gauss, so when we do statistics based on the normal curve, we call it "Gaussian statistics." That is the adventure on which we will now embark. Meanwhile, here are some equations for you to ignore.

This is the shortest way to represent the computations concerning 3 coin flips. "P" is the probability of heads, and "Q" is the probability of tails. They happen to be the same in this case.

2T, 1H=HTT, THT, TTH=3pq^2=3(.5)^3=.375
1T, 2H=HHT, HTH, THH=3p^2q=3(.5)^3=.375

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