1. Key Concepts and Formulas

This problem is a classic application of conditional probability, specifically requiring Bayes' Theorem. The core concepts and formulas are:

• Conditional Probability: The probability of an event $E_1$ occurring given that another event $E_2$ has already occurred is denoted as $P(E_1|E_2)$. It is defined as: $P(E_1|E_2) = \frac{P(E_1 \cap E_2)}{P(E_2)}$ This implies $P(E_1 \cap E_2) = P(E_1|E_2)P(E_2)$, which is the multiplication rule.
• Law of Total Probability: If events $E_1, E_2, \ldots, E_n$ form a partition of the sample space (meaning they are mutually exclusive and their union covers all possibilities), then the probability of any event $F$ can be calculated by summing the probabilities of $F$ occurring through each $E_i$: $P(F) = P(F|E_1)P(E_1) + P(F|E_2)P(E_2) + \ldots + P(F|E_n)P(E_n)$
• Bayes' Theorem: This theorem allows us to "reverse" conditional probabilities. If we know $P(F|E_i)$ (the probability of an observation $F$ given a cause $E_i$) and we want to find $P(E_i|F)$ (the probability of a cause $E_i$ given an observation $F$), Bayes' Theorem states: $P(E_i|F) = \frac{P(F|E_i)P(E_i)}{P(F)}$ By substituting $P(F)$ using the Law of Total Probability, the expanded form is: $P(E_i|F) = \frac{P(F|E_i)P(E_i)}{\sum_{j=1}^{n} P(F|E_j)P(E_j)}$

2. Step-by-Step Solution

Step 1: Define Events and List Given Probabilities To approach the problem systematically, we first define the events and extract the probabilities provided in the question.

• Let $A$ be the event that a randomly selected motorcycle was manufactured at Plant A.
• Let $B$ be the event that a randomly selected motorcycle was manufactured at Plant B.
• Let $S$ be the event that a randomly selected motorcycle is of standard quality.

From the problem statement, we are given:

• Probability of manufacturing at Plant A: $P(A) = 60\% = \frac{60}{100} = \frac{3}{5}$
• Probability of manufacturing at Plant B: Since plants A and B cover the entire production, the events A and B are complementary for the origin of a motorcycle. $P(B) = 1 - P(A) = 1 - \frac{3}{5} = \frac{2}{5}$
• Conditional probability of standard quality given it's from Plant A: $P(S|A) = 80\% = \frac{80}{100} = \frac{4}{5}$
• Conditional probability of standard quality given it's from Plant B: $P(S|B) = 90\% = \frac{90}{100} = \frac{9}{10}$

Our goal is to find $p = P(B|S)$, which is the probability that the motorcycle was manufactured at Plant B, given that it is of standard quality. After finding $p$, we need to calculate $126p$.

Step 2: Calculate the Total Probability of a Motorcycle Being of Standard Quality, $P(S)$ To apply Bayes' Theorem, we first need the overall (unconditional) probability that a randomly picked motorcycle is of standard quality, $P(S)$. We use the Law of Total Probability because a standard quality motorcycle must originate from either Plant A or Plant B.

The Law of Total Probability states: $P(S) = P(S|A)P(A) + P(S|B)P(B)$

Substitute the probabilities obtained in Step 1: $P(S) = \left(\frac{4}{5}\right) \left(\frac{3}{5}\right) + \left(\frac{9}{10}\right) \left(\frac{2}{5}\right)$ $P(S) = \frac{12}{25} + \frac{18}{50}$ To add these fractions, we find a common denominator, which is 50: $P(S) = \frac{12 \times 2}{25 \times 2} + \frac{18}{50}$ $P(S) = \frac{24}{50} + \frac{18}{50}$ $P(S) = \frac{24 + 18}{50} = \frac{42}{50}$ Simplify the fraction: $P(S) = \frac{21}{25}$

Thus, the total probability that a randomly selected motorcycle is of standard quality is $\frac{21}{25}$.

Step 3: Apply Bayes' Theorem to Find $P(B|S)$ We now use Bayes' Theorem to find $p = P(B|S)$, the probability that the motorcycle was manufactured at Plant B given that it is of standard quality.

Bayes' Theorem for this scenario is: $P(B|S) = \frac{P(S|B)P(B)}{P(S)}$

We have all the required values:

• $P(S|B) = \frac{9}{10}$ (from Step 1)
• $P(B) = \frac{2}{5}$ (from Step 1)
• $P(S) = \frac{21}{25}$ (from Step 2)

Substitute these values into the formula: $p = P(B|S) = \frac{\left(\frac{9}{10}\right) \left(\frac{2}{5}\right)}{\frac{21}{25}}$ First, calculate the numerator: $\left(\frac{9}{10}\right) \left(\frac{2}{5}\right) = \frac{18}{50} = \frac{9}{25}$ Now, substitute this back into the expression for $p$: $p = \frac{\frac{9}{25}}{\frac{21}{25}}$ To divide by a fraction, we multiply by its reciprocal: $p = \frac{9}{25} \times \frac{25}{21}$ The 25 in the numerator and denominator cancel out: $p = \frac{9}{21}$ Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 3: $p = \frac{3}{7}$

So, the probability $p$ that the standard quality motorcycle was manufactured at Plant B is $\frac{3}{7}$.

Step 4: Calculate the Final Value $126p$ The problem asks for the value of $126p$. We have found $p = \frac{3}{7}$.

Now, substitute this value into the expression $126p$: $126p = 126 \times \frac{3}{7}$ First, divide 126 by 7: $126 \div 7 = 18$ Then, multiply the result by 3: $18 \times 3 = 54$

Thus, the value of $126p$ is 54.

3. Common Mistakes & Tips

• Misinterpreting Conditional Probabilities: A frequent error is to confuse $P(S|B)$ (probability of standard quality given plant B) with $P(B|S)$ (probability of plant B given standard quality). Bayes' Theorem is specifically designed to bridge this gap. Always clearly identify what is given and what needs to be found.
• Arithmetic Errors with Fractions: Be meticulous when performing calculations with fractions, especially when adding or dividing. Simplifying fractions at each step can help prevent errors and make calculations easier.
• Forgetting the Law of Total Probability: The denominator in Bayes' Theorem, $P(S)$ in this case, represents the overall probability of the observed event. It must be calculated by considering all possible ways the event could occur (i.e., from Plant A or Plant B), using the Law of Total Probability.
• Answering the Wrong Question: Ensure that you provide the final answer requested in the problem ($126p$), not just the intermediate probability $p$.

4. Summary

This problem required us to determine the probability that a motorcycle, known to be of standard quality, originated from Plant B. We began by defining the relevant events and listing all given prior and conditional probabilities. Using the Law of Total Probability, we calculated the overall probability of a motorcycle being of standard quality. Finally, we applied Bayes' Theorem to find the conditional probability that a standard quality motorcycle came from Plant B, denoted as $p$. The calculated value of $p$ was $\frac{3}{7}$, which led to a final answer of $126p = 54$.

5. Final Answer

The final answer is $\boxed{54}$, which corresponds to option (A).