(8) A sequence of n independent experiments is performed....

(8) 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 valid possibilities for the list of outcomes of the experiment are equally likely.

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Key Concepts

Bernoulli Trials

A Bernoulli trial is an experiment with exactly two possible outcomes, commonly labeled as success and failure. Each trial is independent and has a fixed probability of success and failure, forming the basis for modeling binary outcomes in probability theory.

Binomial Distribution

The binomial distribution describes the probability of obtaining a fixed number of successes in a given number of independent Bernoulli trials. It illustrates how outcomes aggregate in repeated, independent experiments with two possible results per trial.

Conditional Probability

Conditional probability is used to determine the probability of an event occurring given that another related event has occurred. In this context, it allows us to analyze the behavior of sequences of outcomes given a fixed number of successes.

Uniform Distribution over Arrangements

When conditioned on the number of successes, every sequence of outcomes with that fixed number of successes is equally likely. This uniformity arises because each specific sequence with the same count of successes has the same product of probabilities, making all such sequences equally probable in the conditional sense.

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