Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
25. Best President In a sample of 1500 adult U.S. citizens, 270 said that Barack Obama was the best president in U.S. history. Two adult U.S. citizens are selected at random.
(Adapted from YouGov)
a. Find the probability that both adult U.S. citizens say that Barack Obama was the best president in U.S. history.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability that two randomly selected adult U.S. citizens both say that Barack Obama was the best president in U.S. history. This involves using the Multiplication Rule for probabilities.
2. Step 2: Calculate the probability of the first person saying Barack Obama was the best president. This is the ratio of the number of people who said Barack Obama was the best president to the total sample size. Mathematically, this is: $\frac{270}{1500}$.
3. Step 3: Calculate the probability of the second person also saying Barack Obama was the best president, assuming the first person has already been selected. Since one person has already been chosen, the total sample size decreases by 1, and the number of people who said Barack Obama was the best president also decreases by 1. This probability is: $\frac{269}{1499}$.
4. Step 4: Apply the Multiplication Rule. The probability of both events occurring (both people saying Barack Obama was the best president) is the product of the two probabilities calculated in Steps 2 and 3. This is: $\frac{270}{1500}\times \frac{269}{1499}$.
5. Step 5: Simplify the expression from Step 4 to find the final probability. This involves multiplying the numerators and denominators of the fractions and simplifying the result if possible.

Key Concepts

Multiplication Rule of Probability

The Multiplication Rule states that the probability of two independent events occurring together is the product of their individual probabilities. In this context, if we want to find the probability that both selected citizens believe Barack Obama is the best president, we first determine the probability for one citizen and then multiply it by itself, as the selections are independent.

Probability Calculation

Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. For this question, the probability that a randomly selected citizen thinks Obama is the best president is 270 out of 1500. This fraction can be simplified to find the probability for one selection, which is then used in the Multiplication Rule.

Independent Events

Independent events are those whose outcomes do not affect each other. In this scenario, the selection of one citizen does not influence the opinion of the second citizen. Understanding that the selections are independent is crucial for applying the Multiplication Rule correctly to find the combined probability.