Problems in Probability

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Fall 2011 Strategic Practice 3: Section 4 (Bernoulli and Binomial) - Question 2
A sequence of n independent experiments is performed. Each experiment is a success with probability p and a failure with probability q = 1 − p. Show that conditional on the number of successes, all possibilities for the list of outcomes of the experiment are equally likely (of course, we only consider lists of outcomes where the number of successes is consistent with the information being conditioned on).
Solution: The probability is 1 / (nCk). This does not depend on a1, ..., an. Thus, for n independent Bernoulli trials, given that there are exactly k successes, the nCk possible sequences consisting of k successes and n − k failures are equally likely. Interestingly, the conditional probability above also does not depend on p (this leads to the notion of a sufficient statistic, which is studied in Stat 111). See iTunes for full solution.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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