Understanding the Problem: Probability of Cake Failure

The question asks us to find the proportion of failed cakes that were baked by cook X. We are given the individual probabilities of failure for each cook (X, Y, and Z) and the proportion of total cakes each cook bakes. This type of problem requires us to use concepts from probability, specifically the Law of Total Probability and Bayes' Theorem.

Given Information

Let's list the probabilities provided in the question:

• Probability that a cake baked by X fails, \(P(F|X) = 0.02\)
• Probability that a cake baked by Y fails, \(P(F|Y) = 0.03\)
• Probability that a cake baked by Z fails, \(P(F|Z) = 0.05\)
• Proportion of cakes baked by X, \(P(X) = 50\% = 0.50\)
• Proportion of cakes baked by Y, \(P(Y) = 30\% = 0.30\)
• Proportion of cakes baked by Z, \(P(Z) = 20\% = 0.20\)

Notice that the proportions of cakes baked by each cook sum up to \(0.50 + 0.30 + 0.20 = 1.00\), covering all cakes baked in the restaurant.

Calculating the Total Probability of Failure

Before we can find the proportion of failures caused by X, we need to determine the overall probability of a cake failing, regardless of who baked it. This is calculated using the Law of Total Probability. The total probability of an event (failure, F) is the sum of the probabilities of that event occurring under each possible condition (being baked by X, Y, or Z), weighted by the probability of each condition.

The formula for the Law of Total Probability is:

\(P(F) = P(F|X)P(X) + P(F|Y)P(Y) + P(F|Z)P(Z)\)

Let's plug in the given values:

\(P(F) = (0.02 \times 0.50) + (0.03 \times 0.30) + (0.05 \times 0.20)\)

Performing the multiplications:

\(P(F) = 0.010 + 0.009 + 0.010\)

Summing these values:

\(P(F) = 0.029\)

So, the total probability of a cake failing is 0.029, or 2.9%.

Calculating the Proportion of Failures Caused by X (Bayes' Theorem)

The question asks for the "proportion of failures caused by X". In probability terms, this is the conditional probability that a cake was baked by X GIVEN that it failed. This is denoted as \(P(X|F)\). We can calculate this using Bayes' Theorem.

Bayes' Theorem states:

\(P(X|F) = \dfrac{P(F|X) \times P(X)}{P(F)}\)

We have already calculated all the components needed for this formula:

• \(P(F|X) = 0.02\)
• \(P(X) = 0.50\)
• \(P(F) = 0.029\)

Now, let's substitute these values into Bayes' Theorem:

\(P(X|F) = \dfrac{0.02 \times 0.50}{0.029}\)

Calculate the numerator:

\(0.02 \times 0.50 = 0.010\)

So, the equation becomes:

\(P(X|F) = \dfrac{0.010}{0.029}\)

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

\(P(X|F) = \dfrac{0.010 \times 1000}{0.029 \times 1000} = \dfrac{10}{29}\)

Therefore, the proportion of failures caused by X is \(\dfrac{10}{29}\).

Summary of Calculations

Total probability of failure, \(P(F) = 0.010 + 0.009 + 0.010 = 0.029\).

Proportion of failures caused by X, \(P(X|F) = \dfrac{\text{Contribution of X to Failure}}{\text{Total Probability of Failure}} = \dfrac{0.010}{0.029} = \dfrac{10}{29}\).

Conclusion

Based on the probabilities provided and applying Bayes' Theorem, the proportion of cake failures that were baked by cook X is \(\dfrac{10}{29}\).

Revision Table: Key Probability Concepts

Additional Information: Applying Bayes' Theorem in Real Life

Bayes' Theorem is a fundamental concept in probability with wide-ranging applications. It's used in many fields to update beliefs or probabilities based on new data or evidence. Here are a few examples:

• Medical Diagnosis: Calculating the probability of a patient having a disease given a positive test result, considering the test's accuracy and the prevalence of the disease.
• Spam Filtering: Determining the probability that an email is spam based on the presence of certain words, using data from previously classified emails.
• Machine Learning: Forming the basis for Naive Bayes classifiers, used in text categorization and other classification tasks.
• Finance: Updating the probability of a stock's price movement based on new economic data or company news.
• Quality Control: Assessing the probability that a defective product came from a specific manufacturing line based on its defect rate and production volume.

In essence, Bayes' Theorem helps us reverse conditional probabilities – going from the probability of an outcome given a cause, to the probability of a cause given an outcome.