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A Bernoulli trial is any expert with a success rate of p, and only 2 outcomes. Probability of k successes in n repeated trials, is given by the binomial distribution.   If we have the r.v. X to measure the #of successes in n trials, then: Pr(X=k)= (nCk) p^k (1-p)^{n-k}   We can see the formula above comes from the fact that the prob of each success is p, so p^k is prob for the k of the n trials that result in success...the (1-p)^{n-k} is the collective prob for each of the n-k failures...then we need to determine how many orderings those n trails (k successes and n-k failures) could of come up in...   This is like ordering the LETTERS....SSSS...FFFFFF, where we have k many S's and n-k many F's...remember the problems with colored flags on the flagpole?  If I have k many white flags to put up into n many slots...and all my other flags are black, how many orderings are there??  nCk...the coefficient in the formula above!   The... read more

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