Problem 71 - Basic Statistics for Business and Economics with Student

Question

A new sports car model has defective brakes 15 percent of the time and a defective steering mechanism 5 percent of the time. Let's assume (and hope) that these problems occur independently. If one or the other of these problems is present, the car is called a "lemon." If both of these problems are present, the car is a "hazard." Your instructor purchased one of these cars yesterday. What is the probability it is: a. A lemon? b. A hazard?

Step-by-Step Solution

Answer

a. 0.1925, b. 0.0075

Step 1: Understand the Problem

We are given probabilities of a sports car having defective brakes and steering mechanism and need to calculate the probabilities of it being a "lemon" (at least one defect) and a "hazard" (both defects).

Step 2: Define the Probabilities

The probability of having defective brakes is 0.15, and the probability of having a defective steering mechanism is 0.05. Since these probabilities are independent, we can define:- \( P(A) = 0.15 \), where \( A \) is the event of defective brakes.- \( P(B) = 0.05 \), where \( B \) is the event of a defective steering mechanism.

Step 3: Calculate Probability of a Hazard

To find the probability of the car being a "hazard," we need both defects present. As the events are independent, use the formula for independent events:\[ P(A \cap B) = P(A) \cdot P(B) = 0.15 \times 0.05 = 0.0075 \]

Step 4: Calculate Probability of a Lemon

A "lemon" is defined when at least one defect is present. Use the formula:\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]Substitute the known values:\[ P(A \cup B) = 0.15 + 0.05 - 0.0075 = 0.1925 \]

Key Concepts

Independent Events

In probability theory, the concept of independent events is key to calculating the likelihood of multiple occurrences. Two events are said to be independent when the occurrence of one event does not affect the occurrence of the second event. In the context of the sports car example, having defective brakes is independent of having a defective steering mechanism.

This means the probability of both events happening simultaneously can be calculated by multiplying their individual probabilities.

• Probability of defective brakes (Event A): 0.15
• Probability of defective steering mechanism (Event B): 0.05

The probability of both the brakes and steering being defective can be calculated using:\[ P(A \cap B) = P(A) \cdot P(B) \]Understanding the independent nature of these defects helps in computing other probabilities like the car being a lemon or a hazard.

Defective Products

Defective products, in this scenario, refer to the sports car having production flaws like bad brakes or faulty steering. These defects have specific probabilities:

• Defective brakes: 15% probability
• Defective steering mechanism: 5% probability

In the world of manufacturing, understanding the probability of defects is crucial. It informs quality control and helps in assessing risks related to the product.

In this case, the defects are considered independent events. Analyzing these defects helps us determine the chances of receiving a car with one or both defects, which affects the customer satisfaction and safety standards.

Event Probability Calculation

Event probability calculation involves figuring out the chance of certain outcomes happening. For our problem, there are two types of events to calculate:

• "Hazard": Both defects are present.
• "Lemon": At least one defect is present.

To calculate the probability of the car being a "hazard" (both defects), use the independent events formula:\[ P(A \cap B) = P(A) \cdot P(B) = 0.15 \times 0.05 = 0.0075 \]This tells us there is a 0.75% chance that both defects are present in the car.

For the "lemon," we apply the formula:\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.15 + 0.05 - 0.0075 = 0.1925 \]This gives a 19.25% chance that at least one of the defects is present.

Compound Probability

Compound probability calculates the likelihood of more than one event occurring in conjunction. It involves scenarios where you are looking at the probability of "either/or" or "and" situations.

In our scenario, we used compound probability for:

• "Hazard": Using "and," finding both defective brakes and steering were calculated through multiplication of independent events probability.
• "Lemon": Using "or," the calculation involved finding at least one defect, hence applying addition and subtraction of overlapping probabilities.

Compound probability is achieved by combining individual probabilities and considering any overlap between events to accurately predict likelihoods.
This method is not only crucial for safety assessments in products like cars but also broadly essential in risk management and decision-making processes across industries.