In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts.

Q: Why can't we just compare the defect rates directly? Machine B has the highest defect rate (4%), so shouldn't it be the most likely culprit?

That is a classic trap! You are looking only at the conditional probability $P(D|B)$. However, Machine B only produces 30% of the bolts, while Machine C produces 50%. You must weigh the defect rate by the production volume. Bayes' Theorem does exactly that by multiplying the production share by the defect rate for each machine.

Q: Is the denominator always the sum of all paths?

Yes, in this context, the denominator represents the 'Total Probability' of the event $D$. Since the machines partition the entire production space (they are mutually exclusive and exhaustive), the sum of all paths leading to $D$ covers every possible way a defective bolt could be created.

Q: What if I get a probability greater than 1?

That is a red flag! Probabilities must always lie between 0 and 1. If you exceed 1, check your arithmetic in the denominator—you likely added an extra term or miscalculated the products. Remember, the numerator is just one part of the total sum in the denominator.

Question

In a bolt factory, machines $A, B$ and $C$ manufacture respectively $20\%, 30\%$ and $50\%$ of the total bolts. Of their output $3, 4$ and $2$ percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective then the probability that it is manufactured by the machine $C$ is

Options:

(A) $\frac{5}{14}$

(B) $\frac{9}{28}$

(C) $\frac{3}{7}$

(D) $\frac{2}{7}$

Sigma Percentile · Accepted Answer

### The Factory Floor: A Journey into Bayes' Theorem

Welcome to the factory floor, where the hum of machinery meets the precision of mathematics. Imagine you are standing in a massive bolt manufacturing facility with three machines—$A$, $B$, and $C$—working tirelessly.

Each machine has a different production capacity and a different rate of error. Today, we are going to solve a mystery: a defective bolt has been found, and we need to determine which machine is responsible.

### Mapping the Factory

To solve this, we first visualize the production process as a tree diagram. Every bolt starts its journey at one of the three machines.

We know the production shares: Machine $A$ produces $20\%$, Machine $B$ produces $30\%$, and Machine $C$ produces $50\%$. Mathematically, we define these as:
$P(A) = 0.2$, $P(B) = 0.3$, and $P(C) = 0.5$.

Notice that these sum to $1$, representing the entire production of the factory. Each machine also has a specific defect rate:
- Machine $A$: $P(D|A) = 0.03$
- Machine $B$: $P(D|B) = 0.04$
- Machine $C$: $P(D|C) = 0.02$

### The Logic of Reverse Probability

We are holding a defective bolt. We know the effect (the defect), but we want to know the cause (the machine). This is the essence of **Bayes' Theorem**.

We are looking for the posterior probability $P(C|D)$, which is the probability that the bolt came from Machine $C$, given that it is defective. Bayes' Theorem provides the bridge:

$$P(C|D) = \frac{P(C) \cdot P(D|C)}{P(D)}$$

The denominator, $P(D)$, represents the total probability of drawing a defective bolt from the entire factory.

### The Calculation

Let us calculate the total probability of a defect, $P(D)$, by summing the probabilities of all paths that lead to a defect:

1. Path $A$: $P(A) \cdot P(D|A) = 0.2 \cdot 0.03 = 0.006$
2. Path $B$: $P(B) \cdot P(D|B) = 0.3 \cdot 0.04 = 0.012$
3. Path $C$: $P(C) \cdot P(D|C) = 0.5 \cdot 0.02 = 0.010$

Adding these together, we find the total probability:

$$P(D) = 0.006 + 0.012 + 0.010 = 0.028$$

Now, we focus on the numerator, which is the probability that the bolt is from Machine $C$ AND is defective:

$$P(C \cap D) = P(C) \cdot P(D|C) = 0.5 \cdot 0.02 = 0.010$$

Finally, we calculate the ratio:

$$P(C|D) = \frac{0.010}{0.028}$$

By multiplying both the numerator and denominator by $1000$, we get $\frac{10}{28}$, which simplifies to the final result of **$\frac{5}{14}$**.

### Conclusion

We have successfully traced the defective bolt back to its source. The probability that it was manufactured by Machine $C$ is **$\frac{5}{14}$**.

Bayes' Theorem is a powerful tool that allows us to update our beliefs based on new evidence. It reminds us that in a world of uncertainty, we can use logic to find the truth hidden behind the data.

Select Answer:

Visualized Solution

Visualizing the Factory Setup

Defining the Events

Machine Production Probabilities

Conditional Defective Rates

Identifying the Goal

The Bayes' Theorem Framework

Calculating the Denominator

Calculating the Numerator

Final Ratio and Simplification

Summary and Conclusion

The Sigma Insight: Bayes' Theorem

The Factory Floor

Mapping the Factory

The Logic of Reverse Probability

The Calculation

Conclusion

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