Table of contents
- 1. Intro to Stats and Collecting Data55m
- 2. Describing Data with Tables and Graphs1h 55m
- 3. Describing Data Numerically1h 45m
- 4. Probability2h 16m
- 5. Binomial Distribution & Discrete Random Variables2h 33m
- 6. Normal Distribution and Continuous Random Variables1h 38m
- 7. Sampling Distributions & Confidence Intervals: Mean1h 3m
- 8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- 9. Hypothesis Testing for One Sample1h 1m
- 10. Hypothesis Testing for Two Samples2h 8m
- 11. Correlation48m
- 12. Regression1h 4m
- 13. Chi-Square Tests & Goodness of Fit1h 20m
- 14. ANOVA1h 0m
4. Probability
Multiplication Rule: Independent Events
Problem 3.2.27c
Textbook Question
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.

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.
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).
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 \( P(\text{none}) = (0.61)^6 \).
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.
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.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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