Textbook Question
Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
20. Coin and Die A coin is tossed and a die is rolled. Find the probability of tossing a tail and then rolling a number greater than 2.
Ch. 3 - Probability
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 3, Problem 3.1.16
Matching Probabilities In Exercises 11-16, match the event with its probability.
a. 0.95
b. 0.005
c. 0.25
d. 0
e. 0.375
f. 0.5
16. You toss a coin four times. What is the probability of tossing tails exactly half of the time?
Verified step by step guidance1
Step 1: Recognize that the problem involves a binomial probability distribution because there are a fixed number of trials (4 coin tosses), two possible outcomes (heads or tails), and the probability of success (tossing tails) is constant at 0.5 for each trial.
Step 2: Use the binomial probability formula to calculate the probability of getting exactly 2 tails in 4 tosses. 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 values into the formula: n = 4 (number of tosses), k = 2 (exactly half of the tosses are tails), and p = 0.5 (probability of tails). The formula becomes: P(X = 2) = (4 choose 2) * (0.5)^2 * (1-0.5)^2.
Step 4: Calculate the binomial coefficient (4 choose 2), which is given by the formula: (n choose k) = n! / [k! * (n-k)!]. For this case, (4 choose 2) = 4! / [2! * (4-2)!] = 6.
Step 5: Multiply the binomial coefficient by the probabilities: P(X = 2) = 6 * (0.5)^2 * (0.5)^2. Simplify the expression to find the final probability.
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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. A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event. In this context, understanding how to calculate the probability of specific outcomes, such as tossing tails in a series of coin flips, is essential.
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Binomial Distribution
The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In the case of tossing a coin, it helps determine the probability of getting a certain number of tails (successes) in a set number of tosses. This concept is crucial for solving the problem of tossing tails exactly half the time in four tosses.
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03:28Mean & Standard Deviation of Binomial Distribution
Combinatorics
Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. In probability problems involving multiple trials, such as coin tosses, combinatorial methods are used to calculate the number of ways a specific outcome can occur. For example, determining how many ways you can get two tails in four tosses is essential for finding the corresponding probability.
Related Practice
Textbook Question
37. Water Pollution An environmental agency is analyzing water samples from 80 lakes for pollution. Five of the lakes have dangerously high levels of dioxin. Six lakes are randomly selected from the sample. Use technology to find how many ways one polluted lake and five nonpolluted lakes can be chosen.
Textbook Question
Using a Frequency Distribution to Find Probabilities In Exercises 49-52, use the frequency distribution at the left, which shows the population of the United States by age group, to find the probability that a U.S. resident chosen at random is in the age range. (Source: U.S. Census Bureau)
52. 65 years old and older
Textbook Question
Warehouse In Exercises 51-54, a warehouse employs 24 workers on first shift, 17 workers on second shift, and 13 workers on third shift. Eight workers are chosen at random to be
interviewed about the work environment.
Find the probability of choosing four third-shift workers.
1
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Textbook Question
80. Unusual Events Can any of the events in Exercises 75-78 be considered unusual? Explain.
Textbook Question
In Exercises 7-14, perform the indicated calculation.
12. (10C7)/(14C7)