Brain Taser
Feb. 17th, 2007 10:40 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
kirisutogomenIf you toss a standard coin in the air, you have a one in two chance of getting heads. The chance of getting heads on two consecutive tosses is one in four; on three consecutive tosses, one in eight; on four consecutive tosses, one in sixteen. Let's assume that you have got four consecutive heads. You are now going to toss again. What are the odds that you will get heads?
The textbook answer is one in two. That's what I would call the mathematician answer.
The scientist answer could be different. Imagine now that you have got not merely four consecutive heads, but forty thousand consecutive heads. Assuming you're not in a Tom Stoppard play, you might start to suspect that maybe this isn't a fair coin. I think a reasonable person (or even a character in a play) would estimate the probability of another head as greater than one in two.
Of course, it doesn't suddenly go from one-half to one at some arbitrary point. Your thought that something fishy is going on grows slowly.
This is (I think) the sort of situation that calls for Bayesian inference. You start with some sort of a priori belief about p, the probability of the coin coming up heads on the next throw. (I'm going to assume that p is constant for a given coin, i.e., tossing it doesn't change the coin. Obviously if I toss a coin forty trillion times, it's going to sustain some wear, which would change p, but Jesus Christ, people, don't be so picky.)
So, if I were 100% convinced that it's a perfectly fair coin, I have a probability of 1 that p=0.5 and a probability of 0 elsewhere. P(p=0.5)=1. Of course, I would have to be a fool to actually believe that. No, more likely I have a probability density curve such that P(0.49<p<0.51)=0.98 or something like that. It's probably not a nice smooth bell curve, because it likely has a couple of spikes at p=0 and p=1.
Then I flip the coin. The first flip comes up heads. Now I have to adjust my estimate of the probability distribution. There isn't a spike at p=0 any more, but the one at p=1 has gotten bigger. The big mound around p=0.5 shifts a little, but hardly enough to notice. Flipping a coin once and getting a head is not terribly good evidence against it being a fair coin.
You see where this is headed. the only other thing I would say is that this is the scientist answer rather than the mathematician answer because it deals in observation, hypothesis, experimentation (or further observation), and adjustment of beliefs based on the further observations.
The textbook answer is one in two. That's what I would call the mathematician answer.
The scientist answer could be different. Imagine now that you have got not merely four consecutive heads, but forty thousand consecutive heads. Assuming you're not in a Tom Stoppard play, you might start to suspect that maybe this isn't a fair coin. I think a reasonable person (or even a character in a play) would estimate the probability of another head as greater than one in two.
Of course, it doesn't suddenly go from one-half to one at some arbitrary point. Your thought that something fishy is going on grows slowly.
This is (I think) the sort of situation that calls for Bayesian inference. You start with some sort of a priori belief about p, the probability of the coin coming up heads on the next throw. (I'm going to assume that p is constant for a given coin, i.e., tossing it doesn't change the coin. Obviously if I toss a coin forty trillion times, it's going to sustain some wear, which would change p, but Jesus Christ, people, don't be so picky.)
So, if I were 100% convinced that it's a perfectly fair coin, I have a probability of 1 that p=0.5 and a probability of 0 elsewhere. P(p=0.5)=1. Of course, I would have to be a fool to actually believe that. No, more likely I have a probability density curve such that P(0.49<p<0.51)=0.98 or something like that. It's probably not a nice smooth bell curve, because it likely has a couple of spikes at p=0 and p=1.
Then I flip the coin. The first flip comes up heads. Now I have to adjust my estimate of the probability distribution. There isn't a spike at p=0 any more, but the one at p=1 has gotten bigger. The big mound around p=0.5 shifts a little, but hardly enough to notice. Flipping a coin once and getting a head is not terribly good evidence against it being a fair coin.
You see where this is headed. the only other thing I would say is that this is the scientist answer rather than the mathematician answer because it deals in observation, hypothesis, experimentation (or further observation), and adjustment of beliefs based on the further observations.
