"Consider a binomial probability distribution with parameters n = 5 and p = 0.2.


a. Construct a binomial probability distribution with these parameters."

Verified step by step guidance

1. Identify the parameters of the binomial distribution: the number of trials \(n = 5\) and the probability of success in each trial \(p = 0.2\).
2. Recall the binomial probability formula for exactly \(k\) successes out of \(n\) trials: \[P(X = k) = \binom{n}{k} p^{k} (1 - p)^{n - k}\] where \(\binom{n}{k}\) is the binomial coefficient representing the number of ways to choose \(k\) successes from \(n\) trials.
3. Calculate the binomial coefficient \(\binom{n}{k}\) for each \(k\) from 0 to 5 using the formula: \[\binom{n}{k} = \frac{n!}{k! (n-k)!}\]
4. For each value of \(k\) (0, 1, 2, 3, 4, 5), substitute \(n\), \(k\), and \(p\) into the binomial probability formula to find \(P(X = k)\): \[P(X = k) = \binom{5}{k} (0.2)^k (0.8)^{5-k}\]
5. List all the probabilities \(P(X = k)\) for \(k = 0, 1, 2, 3, 4, 5\) to construct the full binomial probability distribution.

Key Concepts

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is defined by parameters n (number of trials) and p (probability of success). The distribution gives probabilities for 0 to n successes.

Binomial Probability Formula

The binomial probability formula calculates the probability of exactly k successes in n trials: P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the number of combinations. This formula is essential for constructing the distribution.

Probability Distribution Table

A probability distribution table lists all possible outcomes (0 to n successes) alongside their probabilities. Constructing this table for given n and p helps visualize the distribution and understand the likelihood of each number of successes.