Problem 5 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

In a bulb factory, machines \(A, B\) and \(C\) manufacture \(60 \%, 30 \%\) and \(10 \%\) bulbs, respectively. \(1 \%, 2 \%\) and \(3 \%\) of the bulbs produced, respectively, by \(A, B\) and \(\mathrm{C}\) are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine \(A\). [CBSE-2008]

Step-by-Step Solution

Answer

The probability is 0.4 or 40%.

Step 1: Understand the Known Information

Identify which part of the problem gives us the probabilities of each machine's production share and the defect rates. Machine A produces 60% of the bulbs with a 1% defect rate, Machine B produces 30% with a 2% defect rate, and Machine C produces 10% with a 3% defect rate.

Step 2: Find the Probability of a Defective Bulb

Calculate the total probability of drawing a defective bulb by using the law of total probability. This involves summing the products of the probability of being from a particular machine and the probability of being defective given that it is from that machine:\[P( ext{Defective}) = P( ext{A}) \cdot P( ext{Defective|A}) + P( ext{B}) \cdot P( ext{Defective|B}) + P( ext{C}) \cdot P( ext{Defective|C})=(0.6 \times 0.01) + (0.3 \times 0.02) + (0.1 \times 0.03)\]This results in:\[P( ext{Defective}) = 0.006 + 0.006 + 0.003 = 0.015.\]

Step 3: Use Bayes' Theorem

Find the conditional probability we are interested in, i.e., that the defective bulb came from Machine A. Using Bayes' Theorem:\[P( ext{A|Defective}) = \frac{P( ext{Defective|A}) \cdot P( ext{A})}{P( ext{Defective})}\]Substitute the known values:\[P( ext{A|Defective}) = \frac{0.01 \times 0.6}{0.015}\]This results in:\[P( ext{A|Defective}) = \frac{0.006}{0.015} = 0.4\]

Step 4: Interpret the Result

The probability that the defective bulb was produced by Machine A is 0.4 or 40%. This means that if you pick a defective bulb, there is a 40% chance that it came from Machine A.

Key Concepts

Conditional Probability

Conditional Probability is the probability of an event occurring, given that another event has already taken place. It is a crucial concept in probability theory, helping in situations where the outcomes are not independent.
For calculating conditional probability, Bayes' Theorem is often employed. The theorem links the probability of one event happening based on the probability of another event. For example, if we already know that a bulb is defective, we might want to know the probability that it was manufactured by Machine A. This is represented as \(P(A|\text{Defective})\).
Conditional probability can be calculated with the formula:

• \(P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\)

This formula allows us to figure out unknown probabilities by using known or more easily calculated data. In the bulb factory problem, this was leveraged to find how likely it is that a defective bulb came from a specific machine, given all parts of the exercise.

Law of Total Probability

The Law of Total Probability is a fundamental rule when dealing with probabilities of compound events. It aids in computing the probability of an event by considering all possible scenarios or partitions within the sample space.
In our bulb example, we used the law to determine the overall probability of selecting a defective bulb. This involves summing up the individual probabilities of selecting a defective bulb from each machine.

• First, we calculate the contribution to the defect rate from each machine: \(P(\text{Defective|A}) \cdot P(A)\), \(P(\text{Defective|B}) \cdot P(B)\), and \(P(\text{Defective|C}) \cdot P(C)\).
• Next, add these probabilities together to find the total probability of a defective bulb: \(P(\text{Defective}) = 0.006 + 0.006 + 0.003 = 0.015\).

This method is instrumental in problems involving weighted averages and collective contributions from different events. It ensures that all pathways leading to an outcome are considered.

Probability Theory

Probability Theory encompasses the mathematical framework for quantifying uncertainty and evaluating the likelihood of different outcomes. It serves as the foundation for understanding and solving a wide variety of real-world problems, including those involving complex systems like manufacturing.
In the context of our bulb factory problem, probability theory allows us to:

• Represent the production and defect statistics of each machine as probabilities, making it easier to handle mathematically.
• Use Bayes' Theorem and the Law of Total Probability to compute complex probabilities.
• Interpret and explain results, such as understanding that even though Machine A makes more bulbs, its defect rate still allows us to predict the likelihood of any defect arising from it.

This example shows how probability theory provides tools and principles to manage uncertainty and make informed predictions and decisions based on available data.