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 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

4. Probability

Complements

Statistics

4. Probability

Complements

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2:00 minutes

Problem 4.q.2

Textbook Question

Standard Tests Standard tests, such as the SAT or ACT or MCAT, tend to make extensive use of multiple-choice questions because they are easy to grade using software. If one such multiple choice question has possible correct answers of a, b, c, d, e, what is the probability of a wrong answer if the answer is a random guess?

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Step 1: Understand the problem. The question asks for the probability of a wrong answer when a random guess is made on a multiple-choice question with 5 possible answers (a, b, c, d, e).

Step 2: Recall the formula for probability. The probability of an event occurring is given by the formula: P(Event) = (Number of favorable outcomes) / (Total number of possible outcomes).

Step 3: Identify the total number of possible outcomes. In this case, there are 5 possible answers (a, b, c, d, e), so the total number of possible outcomes is 5.

Step 4: Determine the number of favorable outcomes for a wrong answer. Since only one answer is correct, the number of wrong answers is 5 - 1 = 4.

Step 5: Calculate the probability of a wrong answer using the formula: P(Wrong Answer) = (Number of wrong answers) / (Total number of possible answers) = 4 / 5.

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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 a particular event will occur, expressed as a number between 0 and 1. In the context of multiple-choice questions, it quantifies the chance of selecting a specific answer from a set of possible options. For example, if there are five choices (a, b, c, d, e), the probability of randomly guessing one correct answer is 1 out of 5, or 0.2.

Complementary Events

Complementary events are pairs of outcomes in probability that cover all possible outcomes of an experiment. In this case, the event of guessing correctly (selecting the right answer) and guessing incorrectly (selecting any of the wrong answers) are complementary. If the probability of a correct answer is 0.2, then the probability of a wrong answer is the complement, calculated as 1 - 0.2, which equals 0.8.

Random Guessing

Random guessing refers to making a choice without any prior knowledge or strategy, leading to an equal likelihood of selecting any of the available options. In the scenario of a multiple-choice question with five answers, if a student guesses randomly, each answer has an equal probability of being chosen, which is crucial for calculating the overall probabilities of correct and incorrect answers.

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Textbook Question

Blue Eyes Assume that 35% of us have blue eyes (based on a study by Dr. P. Soria at Indiana University).

b. Find the value of P(B_bar).

Textbook Question

Redundancy in Computer Hard Drives The Seagate ST8000NM0055 hard drive has a 1.22% rate of failures in a year (based on data from Backblaze, Inc.). For the following, assume that all hard drives are that Seagate model.

a. If all of your computer data are stored on a hard disk drive with a copy stored on a second hard disk drive, what is the probability that during a year, you can avoid catastrophe with at least one working drive? Express the result with six decimal places.

Textbook Question

Composite Drug Test Based on the data in Table 4-1, assume that the probability of a randomly selected person testing positive for drug use is 0.126. If drug screening samples are collected from 5 random subjects and combined, find the probability that the combined sample will reveal a positive result. Is that probability low enough so that further testing of the individual samples is rarely necessary?

Textbook Question

Shared Birthdays Find the probability that of 25 randomly selected people, at least 2 share the same birthday.

Textbook Question

In Exercises 1–10, use the data in the accompanying table and express all results in decimal form. (The data are from “The Left-Handed: Their Sinister History,” by Elaine Fowler Costas, Education Resources Information Center, Paper 399519.)

Complement If L represents the event of randomly selecting one of the study subjects and getting someone who writes with their left hand, what does L represent? Find the value of P(L)

Textbook Question

Sampling Eye Color Based on a study by Dr. P. Sorita Soni at Indiana University, assume that eye colors in the United States are distributed as follows: 40% brown, 35% blue, 12% green, 7% gray, 6% hazel.

d. If two people are randomly selected, what is the probability that at least one of them has brown eyes?

Textbook Question

In Exercises 6–10, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from “Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,” by Ohayon et al., Neurology, Vol. 78, No. 20).

Probability Find the probability that at least one of the subjects is a sleepwalker.

Textbook Question

Finding the Probability of the Complement of an Event In Exercises 17-20, the probability that an event will happen is given. Find the probability that the event will not happen.

19. P(E)=0.03

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