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
Basic Concepts of Probability
Problem 14.CRE.1d
Textbook Question
Federal Criminal Defendants The following graph depicts results from a study of defendants in federal crime cases (based on data from the Administrative Office of the U.S. Courts).

d. If three defendants are randomly selected, what is the probability that they all pleaded guilty?

1
Identify the probability of a single defendant pleading guilty from the graph. According to the graph, 90% of defendants pleaded guilty.
Convert the percentage to a probability. Since 90% is equivalent to 0.90, the probability of one defendant pleading guilty is 0.90.
To find the probability that all three defendants pleaded guilty, use the multiplication rule for independent events. The probability of all three defendants pleading guilty is the product of the probabilities of each individual event.
Set up the multiplication of probabilities: \( P(A \text{ and } B \text{ and } C) = P(A) \times P(B) \times P(C) \), where each event (A, B, C) is a defendant pleading guilty.
Substitute the probability of one defendant pleading guilty into the equation: \( P(A \text{ and } B \text{ and } C) = 0.90 \times 0.90 \times 0.90 \). Calculate this product to find the probability that all three defendants pleaded guilty.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability
Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this context, it is used to determine the chance that all three randomly selected defendants pleaded guilty. The probability of a single defendant pleading guilty is 0.90, and for three independent events, the probabilities are multiplied.
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Independent Events
Independent events are those whose outcomes do not affect each other. In this scenario, the selection of one defendant does not influence the outcome of selecting another. Therefore, the probability of all three defendants pleading guilty is the product of the individual probabilities, assuming each selection is independent.
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Multiplication Rule for Independent Events
The multiplication rule for independent events states that the probability of multiple independent events occurring together is the product of their individual probabilities. For this question, the probability that all three defendants pleaded guilty is calculated by multiplying the probability of one defendant pleading guilty (0.90) by itself three times: 0.90 * 0.90 * 0.90.
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