Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
27. Blood Types The probability that a person of Asian descent in the United States has type O+ blood is 39%. At random, six people of Asian descent in the United States are selected. (Source: American National Red Cross)
c. Find the probability that at least one of the six has type O+ blood.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability that at least one of the six randomly selected people of Asian descent has type O+ blood. This is a complementary probability problem, where we will first calculate the probability that none of the six people have type O+ blood and then subtract this value from 1.
2. Step 2: Define the probability of success and failure. The probability that a person has type O+ blood is 0.39 (success), and the probability that a person does not have type O+ blood is 1 - 0.39 = 0.61 (failure).
3. Step 3: Use the Multiplication Rule to calculate the probability that none of the six people have type O+ blood. Since the events are independent, the probability that all six people do not have type O+ blood is given by \( P(\text{none}) = (0.61)^6 \).
4. Step 4: Use the complement rule to find the probability that at least one of the six people has type O+ blood. The complement rule states that \( P(\text{at least one}) = 1 - P(\text{none}) \). Substitute the value of \( P(\text{none}) \) from Step 3 into this formula.
5. Step 5: Simplify the expression \( P(\text{at least one}) = 1 - (0.61)^6 \) to find the final probability. This will give the probability that at least one of the six people has type O+ blood.

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 the probability of one person having type O+ blood is 39%, the probability of not having type O+ blood is 61%. This rule is essential for calculating the probability of multiple independent selections.

Complementary Probability

Complementary probability refers to the likelihood of an event not occurring. For instance, if the probability of a person having type O+ blood is 39%, the complementary probability of not having type O+ blood is 61%. This concept is crucial for solving the problem, as it allows us to find the probability of at least one person having type O+ blood by first calculating the probability that none do.

Binomial Probability

Binomial probability deals with scenarios where there are a fixed number of independent trials, each with two possible outcomes (success or failure). In this case, selecting six people can be modeled as a binomial experiment where 'success' is defined as having type O+ blood. Understanding this concept helps in calculating the overall probability of at least one success in multiple trials.