Binomial Probability Formula. In Exercises 13 and 14, answer the questions designed to help understand the rationale for the binomial probability formula.


Guessing Answers Standard tests, such as the SAT, ACT, or Medical College Admission Test (MCAT), typically use multiple choice questions, each with five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to the first three questions.


c. Based on the preceding results, what is the probability of getting exactly one correct answer when three guesses are made?

Verified step by step guidance

1. Step 1: Recognize that this is a binomial probability problem. The binomial probability formula is given by: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n' is the number of trials, 'k' is the number of successes, 'p' is the probability of success on a single trial, and '1-p' is the probability of failure.
2. Step 2: Identify the values for the problem. Here, 'n' (number of trials) is 3 because there are three questions, 'k' (number of successes) is 1 because we are looking for exactly one correct answer, and 'p' (probability of success) is 1/5 = 0.2 because there is one correct answer out of five choices.
3. Step 3: Calculate the binomial coefficient (n choose k), which is given by the formula: (n choose k) = n! / [k! * (n-k)!]. For this problem, (3 choose 1) = 3! / [1! * (3-1)!] = 3.
4. Step 4: Substitute the values into the binomial probability formula. Using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k), substitute n = 3, k = 1, and p = 0.2. This gives: P(X = 1) = (3 choose 1) * (0.2)^1 * (1-0.2)^(3-1).
5. Step 5: Simplify the expression step by step. First, calculate (3 choose 1), then calculate (0.2)^1, and finally calculate (1-0.2)^2. Multiply these values together to find the probability of getting exactly one correct answer.

Key Concepts

Binomial Probability Distribution

The binomial probability distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, a 'success' is defined as correctly guessing an answer on a multiple-choice test. The formula for calculating the probability of exactly k successes in n trials is given by P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success on a single trial.

Probability of Success and Failure

In the context of guessing answers on a multiple-choice test, the probability of success (correctly guessing an answer) is 1/5, since there is one correct answer out of five options. Conversely, the probability of failure (incorrectly guessing) is 4/5. Understanding these probabilities is crucial for applying the binomial formula to calculate the likelihood of achieving a specific number of correct answers.

Combinatorial Coefficient

The combinatorial coefficient, often represented as 'n choose k' or C(n, k), calculates the number of ways to choose k successes from n trials. This is essential in the binomial probability formula, as it accounts for the different sequences in which successes and failures can occur. For example, if you want to find the probability of getting exactly one correct answer out of three guesses, you need to determine how many different ways this can happen.