Probability Calculation for Defective Bolts

This problem involves calculating conditional probability using Bayes' Theorem. We are given information about the production share of three machines (X, Y, and Z) in a bolt factory and the percentage of defective bolts produced by each machine. We need to find the probability that a randomly selected defective bolt was manufactured by machine X.

Understanding the Given Information

Let's denote the events as follows:

• \(M_X\): A bolt is manufactured by machine X.
• \(M_Y\): A bolt is manufactured by machine Y.
• \(M_Z\): A bolt is manufactured by machine Z.
• \(D\): A bolt is defective.

We are given the following probabilities:

• Percentage of total output from machine X: \(P(M_X) = 25\% = 0.25\)
• Percentage of total output from machine Y: \(P(M_Y) = 35\% = 0.35\)
• Percentage of total output from machine Z: \(P(M_Z) = 40\% = 0.40\)
• Percentage of defective bolts from machine X: \(P(D | M_X) = 2\% = 0.02\)
• Percentage of defective bolts from machine Y: \(P(D | M_Y) = 4\% = 0.04\)
• Percentage of defective bolts from machine Z: \(P(D | M_Z) = 5\% = 0.05\)

We can summarize this information in a table:

Calculating the Total Probability of a Defective Bolt

To find the probability that a randomly drawn bolt is defective, \(P(D)\), we can use the Law of Total Probability. A bolt can be defective and come from machine X, or defective and come from machine Y, or defective and come from machine Z. These are mutually exclusive events.

The probability \(P(D)\) is calculated as:

\(P(D) = P(D | M_X)P(M_X) + P(D | M_Y)P(M_Y) + P(D | M_Z)P(M_Z)\)

Substituting the given values:

\(P(D) = (0.02 \times 0.25) + (0.04 \times 0.35) + (0.05 \times 0.40)\)

\(P(D) = 0.005 + 0.014 + 0.020\)

\(P(D) = 0.039\)

So, the total probability of drawing a defective bolt is 0.039 or 3.9%.

Applying Bayes' Theorem for Conditional Probability

We want to find the probability that the bolt was manufactured by machine X, given that it is defective. This is the conditional probability \(P(M_X | D)\). Bayes' Theorem provides the formula for this:

\(P(M_X | D) = \frac{P(D | M_X)P(M_X)}{P(D)}\)

We have already calculated \(P(D | M_X)\), \(P(M_X)\), and \(P(D)\). Now we substitute these values into the formula:

\(P(M_X | D) = \frac{0.02 \times 0.25}{0.039}\)

\(P(M_X | D) = \frac{0.005}{0.039}\)

To express this as a fraction, we can multiply the numerator and denominator by 1000:

\(P(M_X | D) = \frac{0.005 \times 1000}{0.039 \times 1000} = \frac{5}{39}\)

Conclusion

The probability that a randomly drawn defective bolt was manufactured by machine X is \(\frac{5}{39}\).

Revision Table: Key Probabilities

Additional Information: Bayes' Theorem in Context

Bayes' Theorem is fundamental in probability theory and statistics. It describes how to update the probability of a hypothesis based on new evidence. In this bolt factory example:

• The hypotheses are that the bolt came from Machine X, Y, or Z (\(M_X, M_Y, M_Z\)).
• The initial probabilities of these hypotheses before observing the bolt's condition are \(P(M_X), P(M_Y), P(M_Z)\) (these are called prior probabilities).
• The evidence is that the bolt is defective (\(D\)).
• We know the likelihood of this evidence under each hypothesis (\(P(D|M_X), P(D|M_Y), P(D|M_Z)\)).
• Bayes' Theorem allows us to calculate the updated probability of the hypothesis (e.g., bolt came from X) given the evidence (it's defective). These are called posterior probabilities \(P(M_X|D)\).

The Law of Total Probability is often used in conjunction with Bayes' Theorem to find the denominator, which is the overall probability of the evidence.

This type of problem is common in quality control, medical testing, and other areas where we need to infer the cause (machine) given an effect (defective bolt).