A state senator believes that 25 % of all senators on the...

A state senator believes that $25 \%$ of all senators on the Finance Committee will strongly support the tax proposal she wishes to advance. Suppose that this belief is correct and that 5 senators are approached at random. a. What is the probability that at least 1 of the 5 will strongly support the proposal? b. What is the probability that a majority of the 5 will strongly support the proposal?

A state senator believes that $25 \%$ of all senators on the Finance Committee will strongly support the tax proposal she wishes to advance. Suppose that this belief is correct and that 5 senators are approached at random.
a. What is the probability that at least 1 of the 5 will strongly support the proposal?
b. What is the probability that a majority of the 5 will strongly support the proposal?

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Key Concepts

Binomial Probability Formula

This formula, P(X = k) = C(n, k) · p^k · (1-p)^(n-k), calculates the probability of exactly k successes in n trials. It incorporates the binomial coefficient, which counts the number of ways to choose k successes from n trials, making it essential for determining individual outcome probabilities.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials where each trial has the same probability of success. It is fundamental for analyzing scenarios where outcomes are categorized as success or failure.

Complement Rule

The complement rule states that the probability of an event occurring is 1 minus the probability of it not occurring. This approach is particularly useful for calculating the probability of 'at least one' success by subtracting the probability of zero successes from one.

Cumulative Probability

Cumulative probability involves summing the individual probabilities of multiple outcomes to compute the likelihood of a range of events occurring. This is important when determining the probability of achieving a 'majority' or any collective set of outcomes across several trials.

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