"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)
b. Find the probability that none of the six have type O+ blood."

검증된 단계별 안내

Step 1: Understand the problem. We are tasked with finding the probability that none of the six randomly selected people of Asian descent have type O+ blood. The probability of a person having type O+ blood is given as 39%, or 0.39.

Step 2: Define the complement probability. The complement of a person having type O+ blood is the probability that they do not have type O+ blood. This is calculated as 1 - 0.39 = 0.61.

Step 3: Use the Multiplication Rule. Since the six people are selected independently, the probability that none of them have type O+ blood is the product of the complement probability for each individual. This can be expressed as \( P(\text{none have O+}) = (0.61)^6 \).

Step 4: Write the general formula. The general formula for this type of problem is \( P(\text{none have O+}) = (1 - p)^n \), where \( p \) is the probability of having type O+ blood (0.39 in this case), and \( n \) is the number of people selected (6 in this case).

Step 5: Substitute the values into the formula. Substitute \( p = 0.39 \) and \( n = 6 \) into the formula to calculate \( P(\text{none have O+}) = (1 - 0.39)^6 = (0.61)^6 \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Multiplication Rule

The Multiplication Rule in probability states that the probability of two independent events occurring together is the product of their individual probabilities. In this context, it is used to calculate the likelihood of multiple events happening simultaneously, such as selecting individuals with a specific blood type. For independent events, this rule simplifies the calculation of combined probabilities.

Independent Events

Independent events are those whose outcomes do not affect each other. In the given problem, the selection of one person does not influence the blood type of another selected person. Understanding that the events are independent allows us to apply the Multiplication Rule effectively to find the overall probability of multiple individuals not having type O+ blood.

Complementary Probability

Complementary probability refers to the likelihood of an event not occurring, which is calculated as 1 minus the probability of the event occurring. In this scenario, to find the probability that none of the six selected individuals have type O+ blood, we first determine the probability that one person does not have type O+ blood and then apply the Multiplication Rule to find the combined probability for all six individuals.