Acrophobia USA Today reported results from a survey in which subjects were asked if they are afraid of heights in tall buildings. The results are summarized in the accompanying table. Does this table describe a probability distribution? Why or why not?
Table of contents
- 1. Intro to Stats and Collecting Data1h 14m
- 2. Describing Data with Tables and Graphs1h 56m
- 3. Describing Data Numerically2h 5m
- 4. Probability2h 17m
- 5. Binomial Distribution & Discrete Random Variables3h 6m
- 6. Normal Distribution and Continuous Random Variables2h 11m
- 7. Sampling Distributions & Confidence Intervals: Mean3h 23m
- Sampling Distribution of the Sample Mean and Central Limit Theorem19m
- Distribution of Sample Mean - ExcelBonus23m
- Introduction to Confidence Intervals15m
- Confidence Intervals for Population Mean1h 18m
- Determining the Minimum Sample Size Required12m
- Finding Probabilities and T Critical Values - ExcelBonus28m
- Confidence Intervals for Population Means - ExcelBonus25m
- 8. Sampling Distributions & Confidence Intervals: Proportion2h 10m
- 9. Hypothesis Testing for One Sample5h 8m
- Steps in Hypothesis Testing1h 6m
- Performing Hypothesis Tests: Means1h 4m
- Hypothesis Testing: Means - ExcelBonus42m
- Performing Hypothesis Tests: Proportions37m
- Hypothesis Testing: Proportions - ExcelBonus27m
- Performing Hypothesis Tests: Variance12m
- Critical Values and Rejection Regions28m
- Link Between Confidence Intervals and Hypothesis Testing12m
- Type I & Type II Errors16m
- 10. Hypothesis Testing for Two Samples5h 37m
- Two Proportions1h 13m
- Two Proportions Hypothesis Test - ExcelBonus28m
- Two Means - Unknown, Unequal Variance1h 3m
- Two Means - Unknown Variances Hypothesis Test - ExcelBonus12m
- Two Means - Unknown, Equal Variance15m
- Two Means - Unknown, Equal Variances Hypothesis Test - ExcelBonus9m
- Two Means - Known Variance12m
- Two Means - Sigma Known Hypothesis Test - ExcelBonus21m
- Two Means - Matched Pairs (Dependent Samples)42m
- Matched Pairs Hypothesis Test - ExcelBonus12m
- Two Variances and F Distribution29m
- Two Variances - Graphing CalculatorBonus16m
- 11. Correlation1h 24m
- 12. Regression3h 33m
- Linear Regression & Least Squares Method26m
- Residuals12m
- Coefficient of Determination12m
- Regression Line Equation and Coefficient of Determination - ExcelBonus8m
- Finding Residuals and Creating Residual Plots - ExcelBonus11m
- Inferences for Slope31m
- Enabling Data Analysis ToolpakBonus1m
- Regression Readout of the Data Analysis Toolpak - ExcelBonus21m
- Prediction Intervals13m
- Prediction Intervals - ExcelBonus19m
- Multiple Regression - ExcelBonus29m
- Quadratic Regression15m
- Quadratic Regression - ExcelBonus10m
- 13. Chi-Square Tests & Goodness of Fit2h 21m
- 14. ANOVA2h 29m
5. Binomial Distribution & Discrete Random Variables
Binomial Distribution
Problem 5.2.28b
Textbook Question
In Exercises 25–28, find the probabilities and answer the questions.
Too Young to Tat Based on a Harris poll, among adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. Assume that five adults who regret getting tattoos are randomly selected, and find the indicated probability.
b. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos.
Verified step by step guidance1
Step 1: Recognize that this is a binomial probability problem. The problem involves a fixed number of trials (n = 5), two possible outcomes (success: the adult says they were too young, and failure: the adult does not say they were too young), and a constant probability of success (p = 0.20).
Step 2: Use the binomial probability formula to calculate the probability of exactly one success. The formula is: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k), where 'n' is the number of trials, 'k' is the number of successes, and 'p' is the probability of success.
Step 3: Substitute the given values into the formula. Here, n = 5, k = 1, and p = 0.20. The formula becomes: P(X = 1) = (5 choose 1) * (0.20)^1 * (1 - 0.20)^(5 - 1).
Step 4: Calculate the binomial coefficient (5 choose 1), which is the number of ways to choose 1 success from 5 trials. This is given by the formula: (n choose k) = n! / [k! * (n - k)!]. For (5 choose 1), this simplifies to 5.
Step 5: Combine all the components to express the probability. Multiply the binomial coefficient (5), the probability of success raised to the power of k ((0.20)^1), and the probability of failure raised to the power of (n - k) ((0.80)^4). This gives the final expression for P(X = 1).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Probability
Binomial probability refers to the likelihood of a specific number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the 'success' is defined as an adult stating they were too young to get a tattoo. The formula for calculating this probability is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.
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Probability of Success and Failure
In probability theory, the probability of success is the chance of an event occurring, while the probability of failure is the chance of it not occurring. For this problem, the probability of success (an adult saying they were too young) is 20% (or 0.2), and the probability of failure (not saying they were too young) is 80% (or 0.8). Understanding these probabilities is crucial for calculating the overall binomial probability.
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Combinatorial Coefficient
The combinatorial coefficient, often represented as 'n choose k' or C(n, k), calculates the number of ways to choose k successes from n trials without regard to the order of selection. This is essential in binomial probability calculations, as it accounts for the different combinations of successes and failures. For example, in this scenario, it helps determine how many ways one adult can be identified as saying they were too young among the five selected.
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