Problem 28 - Basic Statistics for Business and Economics with Student

Question

Three defective electric toothbrushes were accidentally shipped to a drugstore by Cleanbrush Products along with 17 nondefective ones. a. What is the probability the first two electric toothbrushes sold will be returned to the drugstore because they are defective? b. What is the probability the first two electric toothbrushes sold will not be defective?

Step-by-Step Solution

Answer

a) The probability is \( \frac{3}{190} \). b) The probability is \( \frac{68}{95} \).

Step 1: Understanding the Scenario

We have a total of 20 electric toothbrushes: 3 are defective and 17 are non-defective. We need to calculate the probability for certain events in two consecutive sales.

Step 2: Calculating Probability for Two Defective Toothbrushes (Part a)

For the first toothbrush, the probability it is defective is \( \frac{3}{20} \). Assuming the first one sold was defective, for the second toothbrush, the probability it is defective is \( \frac{2}{19} \) because one defective toothbrush has already been removed. The combined probability is \( \frac{3}{20} \times \frac{2}{19} = \frac{6}{380} = \frac{3}{190} \).

Step 3: Calculating Probability for Two Non-Defective Toothbrushes (Part b)

For the first non-defective toothbrush, the probability is \( \frac{17}{20} \). Assuming the first one sold was non-defective, the probability the second one is also non-defective is \( \frac{16}{19} \). The combined probability is \( \frac{17}{20} \times \frac{16}{19} = \frac{272}{380} = \frac{136}{190} \).

Step 4: Simplification Check

Check if the fractions \( \frac{3}{190} \) and \( \frac{136}{190} \) are in their simplest terms. \( \frac{3}{190} \) is already simplified, and \( \frac{136}{190} = \frac{68}{95} \) after dividing by 2.

Key Concepts

Defective and Non-defective Items

When we look at a group of items coming from a manufacturer, we often find some items might not work as expected. These are called defective items. Others that work perfectly are non-defective items. Understanding the difference is key when solving statistical problems involving probability. In our scenario, three defective electric toothbrushes were mixed in with 17 non-defective ones, giving us a total of 20. This split helps us see the ratio of items that work to those that don't. This concept is significant not only for identifying problems in a distribution but also for calculating probabilities about what could happen when some of these items are used or sold. This distinction aids in evaluating scenarios such as the probability of picking a defective item first instead of a non-defective one. This forms the basis of calculating more complex probability questions.

Probability Calculation

Probability is a way of measuring how likely an event is to happen. It's calculated by dividing the number of desired outcomes by the total number of possible outcomes. In the context of our toothbrush exercise:- For a defective toothbrush, the probability (\( P_d \)) is the number of defective items divided by the total number, which initially is \( \frac{3}{20} \).- For the second defective toothbrush, the probability changes to \( \frac{2}{19} \) since one defective item has already been removed.By multiplying these probabilities, we calculate the likelihood of two defective toothbrushes being picked one after the other. Similarly, calculating the probability of picking two non-defective toothbrushes follows the same method:- \( \frac{17}{20} \) for the first non-defective.- \( \frac{16}{19} \) for the second, adjusting for the removal of one non-defective.Combining these selective probabilities gives us a clear insight into the likelihood of consecutive picks, whether defective or non-defective.

Statistical Problem Solving

Solving statistical problems involves careful consideration of all possible outcomes and how likely each one is. This allows us to make informed predictions and decisions. In this toothbrush problem, we approach it by first identifying what outcomes we are interested in: 1. **Defective Outcomes:** Both toothbrushes in consecutive sales are defective. 2. **Non-defective Outcomes:** Both toothbrushes in consecutive sales are non-defective. Once these are identified, we use probability calculations to determine how likely these scenarios are. By understanding how to combine probabilities, such as those for sequential events, we can solve more complex statistical problems accurately. This approach lets us break down a situation systematically and ensure every step is based on logical probability operations. With practice, students can apply these calculations to similar real-world situations and statistical queries, gaining confidence in decision-making processes rooted in mathematical reasoning.