Problem 47 - Basic Statistics for Business and Economics with Student

Question

Armco, a manufacturer of traffic light systems, found that under accelerated- life tests, 95 percent of the newly developed systems lasted 3 years before failing to change signals properly. a. If a city purchased four of these systems, what is the probability all four systems would operate properly for at least 3 years? b. Which rule of probability does this illustrate? c. Using letters to represent the four systems, write an equation to show how you arrived at the answer to part (a).

Step-by-Step Solution

Answer

a) Probability is about 0.8145. b) This illustrates the multiplication rule for independent events. c) Equation: \( (0.95)^4 \).

Step 1: Understanding the Problem

The problem involves probability, specifically the probability that certain events occur independently. We're given that the probability of one system lasting at least 3 years is 95%, or 0.95. We need to find the probability that four systems will all last at least 3 years and identify the rule of probability used.

Step 2: Calculating Probability for Multiple Events

To find the probability that all four systems will operate properly for at least 3 years, we use the multiplication rule of independent events. The probability for each system is independent and is 0.95. We need to multiply the probabilities of all four systems working properly: \( P(A) \times P(B) \times P(C) \times P(D) = 0.95 \times 0.95 \times 0.95 \times 0.95 \).

Step 3: Applying the Multiplication Rule

Calculate the probability using the multiplication of independent probabilities: \( P(\text{all four systems working}) = 0.95^4 \). Calculate this using a calculator to find an exact number.

Step 4: Identifying the Probability Rule

The rule used is the multiplication rule for independent events. It states that if two or more events are independent, the probability of all events occurring is the product of their probabilities.

Step 5: Writing the Equation

Using letters to represent the systems A, B, C, and D, the equation for the probability that all systems operate properly is: \( P(A \text{ and } B \text{ and } C \text{ and } D) = 0.95 \times 0.95 \times 0.95 \times 0.95 \) or simply \( (0.95)^4 \).

Key Concepts

Independent Events

When we talk about independent events in probability, we mean that the occurrence or non-occurrence of one event does not affect the probability of another event happening.
For example, in the scenario with the traffic light systems, each system functions independently.
This means that whether one system lasts for three years does not change the likelihood of another system doing the same.

Understanding independent events is critical because it allows us to use the multiplication rule to calculate combined probabilities.
When dealing with such events, remember these key points:

• The outcome of one event does not influence another.
• Each event maintains its own probability, separate from others.

This distinction is important for applying rules like the multiplication rule correctly.

Multiplication Rule

The multiplication rule is a fundamental concept used in probability to calculate the probability of two or more independent events happening simultaneously.
In the context of the traffic light systems, the probability of each system lasting at least 3 years is independent and is 0.95.
To find the probability that all four systems will last for that period, we use the multiplication rule.

Here's how it works:

• If events are independent, the probability of all occurring is found by multiplying their individual probabilities.
• In mathematical language, for events A, B, C, and D, the probability is given by: \( P(A \text{ and } B \text{ and } C \text{ and } D) = P(A) \times P(B) \times P(C) \times P(D) \).
• In our example, it simplifies to: \( 0.95 \times 0.95 \times 0.95 \times 0.95 \), or \( (0.95)^4 \).

This step-by-step calculation helps confirm that the rule is appropriate for determining joint probabilities of independent events.

Probability Calculation

Probability calculation is a critical skill in determining the likelihood of events occurring.
In our problem, each traffic light system has a 95% chance of lasting at least three years.
The task is to find the probability that all four systems meet this criterion.

Here's a simplified guide on how to calculate the probability:

• First, identify the probability of a single event (0.95 for one system).
• Recognize that the events are independent, leading us to use the multiplication rule.
• Combine these probabilities by multiplying them: \( 0.95^4 \), which calculates to approximately 0.8145.
• Thus, the probability that all four systems last at least three years is about 81.45%.

This straightforward calculation indicates how probability concepts can be applied to real-life scenarios effectively.

Accelerated-life Testing

Accelerated-life testing is a method used to estimate the lifespan or durability of a product under intense conditions.
It helps predict how a product will perform under normal conditions by simulating extended use in a short period.
In the traffic systems example, accelerated-life testing helps establish that the probability of a system lasting at least three years is 95%.

Consider these aspects related to accelerated-life testing:

• It's a proactive way to identify potential failures and improve product design.
• Manufacturers can use this data to enhance reliability and inform customers about product lifespan.

Using accelerated-life testing ensures that products meet quality standards and are reliable, providing peace of mind to both manufacturers and consumers.