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 5.1.1

Chapter 5, Problem 5.1.1

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?

Verified step by step guidance

The random variable in this problem is the 'Number of Unlicensed Software Packages' selected in a random trial. A random variable is a numerical outcome of a random phenomenon.

The possible values of the random variable are the numbers listed in the first column of the table: 0, 1, 2, 3, and 4. These represent the count of unlicensed software packages in the sample.

The values of the random variable are numerical because they represent counts, which are quantitative data.

The second column of the table, P(x), represents the probability associated with each value of the random variable. These probabilities must satisfy two conditions: (1) Each probability is between 0 and 1, and (2) the sum of all probabilities equals 1.

To verify the validity of the probability distribution, add all the probabilities in the second column: P(0) + P(1) + P(2) + P(3) + P(4). Ensure the sum equals 1 to confirm it is a valid probability distribution.

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Key Concepts

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

Random Variable

A random variable is a numerical outcome of a random phenomenon. In this context, it represents the number of unlicensed software packages selected from a total of four. Random variables can be discrete, taking on specific values, or continuous, taking on any value within a range.

Probability Distribution

A probability distribution describes how probabilities are assigned to each possible value of a random variable. The table provided shows the probability distribution for the number of unlicensed software packages, indicating the likelihood of each outcome occurring when four packages are selected.

Discrete Values

Discrete values are distinct and separate values that a random variable can take. In this case, the possible values for the number of unlicensed software packages are 0, 1, 2, 3, and 4, all of which are numerical and represent countable outcomes.

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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.

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Textbook Question

In Exercises 25–28, find the probabilities and answer the questions.

Whitus v. Georgia In the classic legal case of Whitus v. Georgia, a jury pool of 90 people was supposed to be randomly selected from a population in which 27% were minorities. Among the 90 people selected, 7 were minorities. Find the probability of getting 7 or fewer minorities if the jury pool was randomly selected. Is the result of 7 minorities significantly low? What does the result suggest about the jury selection process?

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Textbook Question

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Textbook Question

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Textbook Question

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.

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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.

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