4. Probability
Multiplication Rule: Independent Events
1:58 minutesProblem 3.2.28b
Textbook Question
Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
28. Blood Types The probability that a Latinx American person in the United States has type A+ blood is 29%. Four Latinx American people in the United States are selected at random. (Source: American National Red Cross)
b. Find the probability that none of the four have type A+ blood.
Verified step by step guidance1
Step 1: Understand the problem. The probability of a Latinx American person having type A+ blood is given as 29%, or 0.29. The complement of this probability, which is the probability that a person does NOT have type A+ blood, is 1 - 0.29 = 0.71.
Step 2: Identify the scenario. We are selecting four Latinx American people at random and want to find the probability that none of them have type A+ blood. This means all four individuals must fall into the complement category (not having type A+ blood).
Step 3: Apply the Multiplication Rule. Since the events (the blood type of each person) are independent, the probability of all four individuals not having type A+ blood is the product of the probabilities for each individual. This can be expressed as: \( P(\text{none have A+}) = P(\text{not A+}) \times P(\text{not A+}) \times P(\text{not A+}) \times P(\text{not A+}) \).
Step 4: Simplify the expression. Using the complement probability \( P(\text{not A+}) = 0.71 \), the formula becomes: \( P(\text{none have A+}) = 0.71^4 \).
Step 5: Conclude the setup. To find the final probability, calculate \( 0.71^4 \). This will give the probability that none of the four selected individuals have type A+ blood.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Multiplication Rule
The Multiplication Rule in probability states that the probability of two independent events both occurring is the product of their individual probabilities. In this context, it helps calculate the likelihood of multiple events happening together, such as selecting individuals with a specific blood type. For independent events, this rule is essential for determining the combined probability of outcomes.
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Independent Events
Independent events are those whose outcomes do not affect each other. In the given problem, the selection of one Latinx American person does not influence the blood type of another selected person. Understanding independence is crucial for applying the Multiplication Rule correctly, as it allows us to multiply the probabilities of each event without concern for their interactions.
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Complementary Probability
Complementary probability refers to the likelihood of an event not occurring, which can be calculated as 1 minus the probability of the event occurring. In this scenario, to find the probability that none of the four selected individuals have type A+ blood, we first determine the probability of an individual not having type A+ blood and then apply the Multiplication Rule to find the overall probability for all four individuals.
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