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.


a. Use the multiplication rule to find the probability that the first two guesses are wrong and the third is correct. That is, find P(WWC), where W denotes a wrong answer and C denotes a correct answer.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability of guessing wrong on the first two questions and guessing correctly on the third question. This is represented as P(WWC), where W denotes a wrong answer and C denotes a correct answer.
2. Step 2: Recall the multiplication rule for independent events. The probability of multiple independent events occurring together is the product of their individual probabilities. In this case, the events are guessing wrong on the first question, guessing wrong on the second question, and guessing correctly on the third question.
3. Step 3: Determine the probability of each event. Since there are 5 possible answers for each question and only 1 correct answer, the probability of guessing wrong (W) is 4/5, and the probability of guessing correctly (C) is 1/5.
4. Step 4: Apply the multiplication rule. Multiply the probabilities of the three events: P(WWC) = P(W) × P(W) × P(C). Substitute the probabilities: P(WWC) = (4/5) × (4/5) × (1/5).
5. Step 5: Simplify the expression. Multiply the fractions to find the final probability. The result will represent the probability of guessing wrong on the first two questions and correctly on the third question.

Key Concepts

Binomial Probability

The binomial probability formula calculates the likelihood of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, a 'success' could be guessing the correct answer, while a 'failure' would be guessing incorrectly. The 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, and p is the probability of success.

Multiplication Rule

The multiplication rule in probability states that the probability of two independent events occurring together is the product of their individual probabilities. For example, if the probability of guessing incorrectly on the first question is 4/5 and the probability of guessing correctly on the third question is 1/5, the overall probability of the sequence of events (WWC) can be calculated by multiplying these probabilities together.

Independent Events

Independent events are those whose outcomes do not affect each other. In the context of guessing answers on a multiple-choice test, each guess is independent because the outcome of one guess does not influence the others. This independence allows us to apply the multiplication rule to find the probability of a specific sequence of correct and incorrect answers.