[math-fun] Coin-tossing
Maybe I've misunderstood the question, but I'm wondering if JP is asking for something that cannot exist. Suppose a 'coin', simulated for example by a random-number generator on a computer, has a probability 'p' of coming up Heads. Then the probability of getting 'X' Heads out of 'n' is defined by a binomial formula involving 'p', 'X' and 'n'. However, if I observe 'X' heads occurring in 'n' tosses of the coin, I have a different piece of information - in fact less information. Even if X/n = p, I don't know that p is the probability of coming up Heads. So, in terms of knowledge, probability and statistics, I'm 'in a different place'. The case of getting HH on two coin tosses (with probability p^2) shows this. The case of getting H on one coin toss shows this too. If I could derive a binomial formula for the number of Heads in n' tosses of the coin, it would be equivalent to knowing what the probability of Heads is on each toss. But I don't know this, so I can't derive a binomial formula from observing a sequence of coin-tosses. Guy
Let me answer the question in the case n=2, n'=1. If I use the random permutation strategy we've discussed (to compute Y by making use of X and some extra randomness), then P(Y=1) = P(X=2)*1 + P(X=1)*1/2 + P(X=0)*0 = p^2 + 2p(1-p)/2 + 0 = p^2 + p-p^2 = p, so Y is governed by the Binomial(p,1) distribution. Jim On 6/24/12, Guy Haworth <g.haworth@reading.ac.uk> wrote:
Maybe I've misunderstood the question, but I'm wondering if JP is asking for something that cannot exist.
Suppose a 'coin', simulated for example by a random-number generator on a computer, has a probability 'p' of coming up Heads.
Then the probability of getting 'X' Heads out of 'n' is defined by a binomial formula involving 'p', 'X' and 'n'.
However, if I observe 'X' heads occurring in 'n' tosses of the coin, I have a different piece of information - in fact less information.
Even if X/n = p, I don't know that p is the probability of coming up Heads. So, in terms of knowledge, probability and statistics, I'm 'in a different place'.
The case of getting HH on two coin tosses (with probability p^2) shows this. The case of getting H on one coin toss shows this too.
If I could derive a binomial formula for the number of Heads in n' tosses of the coin, it would be equivalent to knowing what the probability of Heads is on each toss.
But I don't know this, so I can't derive a binomial formula from observing a sequence of coin-tosses.
Guy
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I wrote "Y", but I think to be consistent with earlier notations I should've written "X'", I think. Anyway, a point I should stress is that being able to sample from Binomial(p,n') is not the same thing as knowing (or being able to infer) the exact value of p. (Indeed, as an extreme case, tossing a coin of unknown bias and announcing the result does not require knowing what the bias is.) Jim On 6/24/12, James Propp <jamespropp@gmail.com> wrote:
Let me answer the question in the case n=2, n'=1.
If I use the random permutation strategy we've discussed (to compute Y by making use of X and some extra randomness), then P(Y=1) = P(X=2)*1 + P(X=1)*1/2 + P(X=0)*0 = p^2 + 2p(1-p)/2 + 0 = p^2 + p-p^2 = p, so Y is governed by the Binomial(p,1) distribution.
Jim
On 6/24/12, Guy Haworth <g.haworth@reading.ac.uk> wrote:
Maybe I've misunderstood the question, but I'm wondering if JP is asking for something that cannot exist.
Suppose a 'coin', simulated for example by a random-number generator on a computer, has a probability 'p' of coming up Heads.
Then the probability of getting 'X' Heads out of 'n' is defined by a binomial formula involving 'p', 'X' and 'n'.
However, if I observe 'X' heads occurring in 'n' tosses of the coin, I have a different piece of information - in fact less information.
Even if X/n = p, I don't know that p is the probability of coming up Heads. So, in terms of knowledge, probability and statistics, I'm 'in a different place'.
The case of getting HH on two coin tosses (with probability p^2) shows this. The case of getting H on one coin toss shows this too.
If I could derive a binomial formula for the number of Heads in n' tosses of the coin, it would be equivalent to knowing what the probability of Heads is on each toss.
But I don't know this, so I can't derive a binomial formula from observing a sequence of coin-tosses.
Guy
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