This problem involves calculating a conditional probability using Bayes' Theorem. We are given information about two shock absorber suppliers, X and Y, and the reliability rates of their products after a quality test. Our goal is to find the probability that a randomly chosen reliable shock absorber was supplied by Y.

Shock Absorber Problem Overview

An automobile plant sources shock absorbers from two primary suppliers: X and Y. Each supplier contributes a specific percentage of the total shock absorbers. All products undergo a quality test, and those that pass are deemed reliable. We have information about the individual reliability rates from each supplier. We need to determine the likelihood that a reliable shock absorber came from supplier Y.

Supplier Probabilities and Reliability Rates

Let's define the events and their given probabilities:

• Let \(X\) be the event that a shock absorber is supplied by X.
• Let \(Y\) be the event that a shock absorber is supplied by Y.
• Let \(R\) be the event that a shock absorber is found to be reliable.

From the problem statement, we have the following initial probabilities:

• Probability of a shock absorber being from supplier X, \(P(X) = 60\% = 0.60\).
• Probability of a shock absorber being from supplier Y, \(P(Y) = 40\% = 0.40\).
• Probability that a shock absorber is reliable, given it's from supplier X, \(P(R|X) = 96\% = 0.96\).
• Probability that a shock absorber is reliable, given it's from supplier Y, \(P(R|Y) = 72\% = 0.72\).

Bayes' Theorem for Conditional Probability

We are asked to find the probability that a randomly chosen shock absorber, which is found to be reliable, is made by Y. This is a conditional probability, specifically \(P(Y|R)\). Bayes' Theorem is the appropriate tool for this calculation. Bayes' Theorem states:

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

In our context, to find \(P(Y|R)\), the formula becomes:

\(P(Y|R) = \frac{P(R|Y) \cdot P(Y)}{P(R)}\)

Before we can apply this directly, we first need to calculate the overall probability that a randomly chosen shock absorber is reliable, \(P(R)\).

Total Reliability Probability Calculation

The total probability of a shock absorber being reliable, \(P(R)\), can be found using the Law of Total Probability. This law considers all possible ways for the event to occur. In this case, a shock absorber can be reliable if it comes from supplier X and is reliable, OR if it comes from supplier Y and is reliable.

\(P(R) = P(R|X) \cdot P(X) + P(R|Y) \cdot P(Y)\)

Let's substitute the known values into this equation:

\(P(R) = (0.96 \cdot 0.60) + (0.72 \cdot 0.40)\)

Now, perform the multiplications:

• \(0.96 \cdot 0.60 = 0.576\) (This is the probability of a reliable shock absorber coming from X)
• \(0.72 \cdot 0.40 = 0.288\) (This is the probability of a reliable shock absorber coming from Y)

Add these two values to find the total probability of a reliable shock absorber:

\(P(R) = 0.576 + 0.288 = 0.864\)

So, the overall probability that a randomly chosen shock absorber is reliable is \(0.864\).

Supplier Y Conditional Probability

Now that we have \(P(R)\), we can use Bayes' Theorem to calculate \(P(Y|R)\), the probability that a reliable shock absorber was made by supplier Y.

\(P(Y|R) = \frac{P(R|Y) \cdot P(Y)}{P(R)}\)

Substitute the values we have:

\(P(Y|R) = \frac{0.72 \cdot 0.40}{0.864}\)

Calculate the numerator:

\(0.72 \cdot 0.40 = 0.288\)

Now, divide the numerator by the denominator:

\(P(Y|R) = \frac{0.288}{0.864}\)

\(P(Y|R) \approx 0.3333...\)

Rounding this value to three decimal places, we get \(0.334\).

Therefore, the probability that a randomly chosen shock absorber, which is found to be reliable, is made by Y is approximately \(0.334\).