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scott wrote:
> If a weighted coin is tossed N times, count the times it lands heads as
> H. The probability of landing heads is P(H).
I presume by this you explicitly mean that P(H) /= 50%?
> I assume the expected value of H is N*P(H).
Yes. That would be the definition of "probability". (Assuming all the
trails are independent.)
> But how do I find out what range of H is X% likely to happen? For
> example, I want to be able to say that in 90% of cases H will be in the
> range (A,B).
Now you're talking about probability distributions. If you can decide
what probability distribution this experiment has, you can [potentially]
directly compute the result you're after.
Wikipedia informs me that what you're looking for is a "binomial
distribution". According to this, we have the probability of getting
exactly K heads as
(N choose K) * P(H)^K * (1 - P(H))^(N-K)
More usefully for you, the cumulative probability - the probability of K
heads OR LESS, is given by
I[1 - P(H)] (N - K, N + K)
where I[x](a, b) represents the regularised incomplete beta function,
http://en.wikipedia.org/wiki/Regularized_incomplete_beta_function
Unless Excel has a built-in function to compute I(), I would suggest it
would be simpler and easier to directly compute the sum over the
individual probabilities. (Especially since you want the probability for
a value between K1 and K2, not between K1 and 0.)
You can also compute the cumulative probability by the integral
(N - K) (N choose K) integral[0 .. 1-P(H)] t^(N-K-1) (1 - t)^K dt
But I'm guessing Excel can't do that.
Apparently you can also estimate the maximum probability by
exp( -2 * (N * P(H) - K)^2 / N )
(The true cumulative probability is less than or equal to this.
Apparently for P(H) = 50% it's quite a good approximation, but I'm not
sure for other values.)
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