Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

Problem 4.1.25

Chapter 4, Problem 4.1.25

Determining a Missing Probability In Exercises 25 and 26, determine the missing probability for the probability distribution.

Verified step by step guidance

Step 1: Recall that for a probability distribution, the sum of all probabilities must equal 1. This is a fundamental property of probability distributions.

Step 2: Write the equation to represent the sum of probabilities: $ P(0) + P(1) + P(2) + P(3) + P(4) = 1 $. Substitute the known values: $ 0.06 + 0.12 + 0.18 + P(3) + 0.30 = 1 $.

Step 3: Combine the known probabilities: $ 0.06 + 0.12 + 0.18 + 0.30 = 0.66 $. This simplifies the equation to $ 0.66 + P(3) = 1 $.

Step 4: Solve for $ P(3) $ by subtracting $ 0.66 $ from both sides: $ P(3) = 1 - 0.66 $.

Step 5: Verify that the sum of all probabilities equals 1 after finding $ P(3) $. This ensures the distribution is valid.

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Key Concepts

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

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. For discrete random variables, it lists each possible value and its corresponding probability, ensuring that the sum of all probabilities equals 1. Understanding this concept is crucial for determining any missing probabilities in a given distribution.

Sum of Probabilities

In a valid probability distribution, the sum of all probabilities must equal 1. This principle is essential for solving problems involving missing probabilities, as it allows us to set up an equation where the unknown probability can be calculated by subtracting the sum of known probabilities from 1.

Complementary Events

Complementary events are pairs of outcomes in a probability space that cover all possible outcomes. In the context of probability distributions, if we know the probabilities of certain outcomes, we can find the probability of the missing outcome by recognizing that it is the complement of the sum of the known probabilities. This concept aids in efficiently determining missing values.

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Complementary Events

Related Practice

Textbook Question

Multinomial Experiments In Exercises 39 and 40, use the information below.

A multinomial experiment satisfies these conditions.

The experiment has a fixed number of trials n, where each trial is independent of the other trials.

Each trial has k possible mutually exclusive outcomes:

Each outcome has a fixed probability. So, . The sum of the probabilities for all outcomes is

The number of times occurs is , the number of times occurs is , the number of times occurs is , and so on.

The discrete random variable x counts the number of times that each outcome occurs in n independent trials where . The probability that x will occur is

Genetics According to a theory in genetics, when tall and colorful plants are crossed with short and colorless plants, four types of plants will result: tall and colorful, tall and colorless, short and colorful, and short and colorless, with corresponding probabilities of , and . Ten plants are selected. Find the probability that 5 will be tall and colorful, 2 will be tall and colorless, 2 will be short and colorful, and 1 will be short and colorless.

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Textbook Question

"Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Typographical Errors A newspaper finds that the mean number of typographical errors per page is four. Find the probability that the number of typographical errors found on any given page is (a) exactly three, (b) at most three, and (c) more than three."

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Textbook Question

Geometric Distribution: Mean and Variance In Exercises 29 and 30, use the fact that the mean of a geometric distribution is μ = 1/p and the variance is

sigma^2 = q/p^2

Paycheck Errors A company assumes that 0.5% of the paychecks for a year were calculated incorrectly. The company has 200 employees and examines the payroll records from one month. (a) Find the mean, variance, and standard deviation. (b) How many employee payroll records would you expect to examine before finding one with an error?

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Textbook Question

Finding an Expected Value In Exercises 37 and 38, find the expected value E(x) to the player for one play of the game. If x is the gain to a player in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose.

A high school basketball team is selling \$10 raffle tickets as part of a fund-raising program. The first prize is a trip to the Bahamas valued at \$5460, and the second prize is a weekend ski package valued at \$496. The remaining 18 prizes are \$100 gas cards. The number of tickets sold is 3500.

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Textbook Question

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Textbook Question

Finding Binomial Probabilities In Exercises 19–26, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B.

Civil Rights Fifty-nine percent of U.S. adults think that civil rights for Black Americans have improved during their lifetime. You randomly select seven U.S. adults. Find the probability that the number who think that civil rights for Black Americans have improved during their lifetime is (a) exactly one and (b) exactly five. (Source: Gallup)

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