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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblema 5.R.1
Capitolo 5, Problema 5.R.1

In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.


Workplace Drug Testing Find the probability that exactly two of the ten workers test positive for illegal drugs.

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Step 1: Recognize that this is a binomial probability problem. The binomial distribution is used because there are a fixed number of trials (n = 10 workers), each trial has two possible outcomes (positive or negative drug test), the probability of success (testing positive) is constant (p = 0.042), and the trials are independent.
Step 2: Write the formula for the binomial probability: 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, 'p' is the probability of success, and '1 - p' is the probability of failure.
Step 3: Substitute the given values into the formula. Here, n = 10, k = 2, and p = 0.042. The formula becomes: P(X = 2) = (10 choose 2) * (0.042)^2 * (1 - 0.042)^(10 - 2).
Step 4: Calculate the binomial coefficient (10 choose 2), which is given by the formula: (n choose k) = n! / [k! * (n - k)!]. For this problem, (10 choose 2) = 10! / [2! * (10 - 2)!].
Step 5: Simplify the remaining terms: (0.042)^2 represents the probability of two workers testing positive, and (1 - 0.042)^8 represents the probability of the remaining eight workers testing negative. Multiply all these components together to find the probability.

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Binomial Probability

Binomial probability refers to the probability of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, testing positive for illegal drugs is considered a 'success,' and the scenario involves ten independent tests, making it suitable for binomial distribution analysis.
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Calculating Probabilities in a Binomial Distribution

Probability of Success and Failure

In a binomial distribution, the probability of success (testing positive) is denoted as 'p,' while the probability of failure (testing negative) is 'q,' where q = 1 - p. For this problem, p is 0.042 (4.2%), and q is 0.958 (95.8%). Understanding these probabilities is crucial for calculating the likelihood of different outcomes.
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Introduction to Probability

Combination Formula

The combination formula, denoted as C(n, k) or nCk, calculates the number of ways to choose k successes from n trials without regard to the order of selection. This is essential for determining how many different groups of two workers can test positive out of ten, which is a key component in calculating the overall probability of exactly two positives.
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Combinations
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In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.


Workplace Drug Testing Find the mean and standard deviation for the numbers of workers in groups of ten who test positive for illegal drugs.

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In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.


Workplace Drug Testing If none of the ten workers tests positive for illegal drugs, is that a significantly low result?

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