A random committee of size 3 is selected from 4 doctors and...

A random committee of size 3 is selected from 4 doctors and 2 nurses. Write a formula for the probability distribution of the random variable $X$ representing the number of doctors on the committee. Find $P(2 \leq X \leq 3)$.

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Key Concepts

Probability Distribution

A probability distribution assigns probabilities to the possible outcomes of a random variable. It provides a complete description of the likelihood of various outcomes in an experiment based on combinatorial calculations, thereby allowing one to compute probabilities for events defined over the range of the random variable.

Hypergeometric Distribution

The hypergeometric distribution models the probability of drawing a specific number of successes from a finite population in situations where samples are drawn without replacement. It involves counting the ways to obtain a given number of successful outcomes and dividing by the total number of possible outcomes, making it particularly relevant in scenarios where the composition of the sample changes as items are selected.

Combinations

Combinations are a fundamental counting technique used to determine the number of ways to select items from a set when the order of selection does not matter. They are calculated using the binomial coefficient formula, which is essential for evaluating probabilities in problems involving random sampling without replacement.

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