Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.1.17

Chapter 5, Problem 5.1.17

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).

Range Rule of Thumb for Significant Events Use the range rule of thumb to determine whether 4 matches is a significantly high number of matches.

Verified step by step guidance

Step 1: Understand the problem. The goal is to determine whether 4 matches is a significantly high number of matches using the range rule of thumb. The range rule of thumb states that values are considered significantly high if they are greater than the mean plus two standard deviations.

Step 2: Calculate the mean (μ) of the random variable x. Use the formula for the mean of a probability distribution: μ = Σ[x * P(x)], where x represents the number of matching digits and P(x) represents the corresponding probability.

Step 3: Calculate the standard deviation (σ) of the random variable x. Use the formula for the standard deviation of a probability distribution: σ = √Σ[(x - μ)^2 * P(x)], where μ is the mean calculated in Step 2.

Step 4: Apply the range rule of thumb. Compute the threshold for significantly high values using the formula: μ + 2σ. Compare the value of 4 matches to this threshold.

Step 5: Interpret the result. If 4 matches exceed the threshold calculated in Step 4, it is considered a significantly high number of matches. Otherwise, it is not considered significant.

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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 of a random variable are distributed across its possible values. In this case, the table shows the probabilities of matching digits in the California Daily 4 lottery, where the random variable x represents the number of digits that match. Understanding this distribution is crucial for analyzing the likelihood of different outcomes in the lottery.

Random Variable

A random variable is a numerical outcome of a random phenomenon. In the context of the lottery question, the random variable x represents the number of digits that match the drawn numbers in the same order. Recognizing how random variables function helps in calculating probabilities and understanding the nature of the outcomes in probabilistic scenarios.

Range Rule of Thumb

The Range Rule of Thumb is a statistical guideline used to determine whether a particular outcome is significantly high or low compared to the expected range of values. It suggests that if an outcome is more than two standard deviations away from the mean, it can be considered significant. In this lottery context, applying this rule will help assess whether matching all four digits is an unusually high event.

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Textbook Question

If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has A objects of one type (such as lottery numbers you selected), while the remaining B objects are of the other type (such as lottery numbers you didn’t select), and if n objects are sampled without replacement (such as six drawn lottery numbers), then the probability of getting x objects of type A and objects of type B is

In New Jersey’s Pick 6 lottery game, a bettor selects six numbers from 1 to 49 (without repetition), and a winning six-number combination is later randomly selected. Find the probability of getting exactly four winning numbers with one ticket.

Textbook Question

Independent Events Again assume that when randomly selecting a speaking character in a movie, the probability of getting a female is 0.331, as in Exercise 1. If we want to find the probability of 20 females when 50 different speaking characters are randomly selected from a population of 1500 speaking characters, are the 50 selections independent? Using the 5% guideline for cumbersome calculations, can they be treated as being independent?

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Textbook Question

Biometric Security In a USA Today survey of 510 people, 270 (or 53%) said that we should replace passwords with biometric security, such as fingerprints. Use the following probabilities related to determining whether the result of 270 is significantly high (assuming the true rate is 50%). Is 270 significantly high? What should be concluded about the claim that the majority of the population says that we should replace passwords with biometric security? Explain.

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Textbook Question

Random Variable The accompanying table lists probabilities for the corresponding numbers of unlicensed software packages when four software packages are randomly selected in China. What is the random variable, what are its possible values, and are its values numerical?

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Textbook Question

Discrete or Continuous? Is the random variable given in the table from Exercise 1 discrete or continuous? Explain.

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Textbook Question

40% of consumers believe that cash will be obsolete in the next 20 years (based on a survey by J.P. Morgan Chase). In each of Exercises 15–20, assume that 8 consumers are randomly selected. Find the indicated probability.

Find the probability that no more than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.

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