Suppose that n independent trials, each of which results in...

Suppose that $n$ independent trials, each of which results in any of the outcomes 0,1 , or 2 with respective probabilities, $p_{0}, p_{1}$, and $p_{2}, \sum_{i=0}^{2} p_{i}=1$, are performed. Find the probability that outcomes 1 and 2 both occur at least once.

Key Concepts

Independent Trials

This concept refers to performing experiments or trials in such a way that the outcome of one trial does not influence or affect the outcome of any other trial. In probabilistic experiments, independence allows the joint probability of a series of events to be calculated as the product of the individual event probabilities.

Complement Rule

This rule is used to determine the probability of an event by subtracting the probability of its complement (the event not occurring) from one. It is particularly useful when it is easier to calculate the probability that an event does not happen, especially in cases involving 'at least one' occurrences.

Inclusion-Exclusion Principle

A core counting method in probability theory, this principle helps calculate the probability of the union of multiple events by adding the probabilities of individual events and then subtracting the probabilities of their intersections to correct for over-counting. It is especially useful when determining the probability that at least one of several events occurs.

Multinomial Experiments

These experiments involve trials where each trial can result in more than two possible outcomes, with fixed probabilities for each outcome. This framework generalizes the simpler binomial experiments and provides a way to study probabilities when there are multiple distinct outcomes in each independent trial.

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