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Chapter 3: Problem 38

Three cooks, \(A, B\), and \(C\), bake a special kind of cake, and with respective probabilities \(.02, .03\), and \(.05\) it fails to rise. In the restaurant where they work, \(A\) bakes 50 percent of these cakes, \(B 30\) percent, and \(C 20\) percent. What proportion of "failures" is caused by \(A ?\)

Short Answer

Expert verified
The proportion of failures caused by Cook \(A\) is approximately \(34.48\%\).

Step by step solution

01

Identify What We Know

We're given the following information: - Cook \(A\) has a failure rate of \(.02\) and bakes \(50\%\) of the cakes. - Cook \(B\) has a failure rate of \(.03\) and bakes \(30\%\) of the cakes. - Cook \(C\) has a failure rate of \(.05\) and bakes \(20\%\) of the cakes. We want to find the proportion of failures caused by Cook \(A\): \(P(A | \text{Failed})\).
02

Calculate the Individual Probabilities of Failure

First, we will calculate the individual probabilities of each cook causing a failure, using the given failure rate and the percentage of cakes they bake. For Cook \(A\): \(P(A \cap \text{Failed}) = P(\text{Failed} | A) \times P(A) = 0.02 \times 0.50 = 0.01\) For Cook \(B\): \(P(B \cap \text{Failed}) = P(\text{Failed} | B) \times P(B) = 0.03 \times 0.30 = 0.009\) For Cook \(C\): \(P(C \cap \text{Failed}) = P(\text{Failed} | C) \times P(C) = 0.05 \times 0.20 = 0.01\)
03

Calculate the Total Probability of Failure

Now, we will calculate the total probability of a cake failing to rise: \(P(\text{Failed}) = P(A \cap \text{Failed}) + P(B \cap \text{Failed}) + P(C \cap \text{Failed}) = 0.01 + 0.009 + 0.01 = 0.029\)
04

Calculate the Proportion of Failures Caused by Cook A

Finally, we will use the formula for conditional probability to find the proportion of failed cakes caused by Cook \(A\): \(P(A | \text{Failed}) = \frac{P(A \cap \text{Failed})}{P(\text{Failed})} = \frac{0.01}{0.029} \approx 0.3448\) Therefore, about \(34.48\%\) of the failed cakes are caused by Cook \(A\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability of Failure
Understanding the probability of failure is important when assessing the risk associated with any event, process, or activity. This concept refers to the likelihood that a system or component will not perform its intended function. In our exercise involving the three cooks, A, B, and C, and their cake-baking, the failure refers to the cake not rising as expected. Each cook has a different failure rate, which is a numerical value representing the probability that their cake will fail to rise.

As an example, Cook A has a failure rate of 0.02, meaning there's a 2% chance that any given cake they bake will not rise. These probabilities are crucial when we start to look into the bigger picture of how many failures can be expected in total from the restaurant's cake-baking operations. By understanding the individual failure rates, we can calculate the overall impact each cook has on the probability of cake failures in the restaurant.
Total Probability Rule
The total probability rule is a fundamental principle in probability theory. It states that the total probability of an event can be found by considering all the different ways that event can occur. In mathematical terms, if a set of events covers all possible outcomes, the probability of a given event is the sum of the probabilities of that event occurring in conjunction with each of the possibilities.

In the context of our cake-baking problem, we use this rule to calculate the total probability of a cake failing to rise, regardless of which cook bakes it. This is done by adding together the individual probabilities of failure for each cook, taking into account both their failure rates and their proportion of the total cakes baked. For instance, Cook C contributes to the total failure probability not just by their higher failure rate but also based on how many cakes they bake.
Proportion of Outcomes
The notion of the proportion of outcomes deals with the relative frequency of a specific outcome within a set of all possible outcomes. It is essential when trying to understand the contribution of a particular element to the overall results. Proportion is fundamentally about comparing parts to the whole and is often expressed in terms of percentage.

With our cooks and their cakes, we seek the proportion of failures directly attributable to Cook A. After finding the total failure probability, we then use conditional probability to find the percentage of the time a cake failure can be traced back to Cook A. This is a crucial step for the restaurant's management if they wish to reduce the number of unsuccessful cakes and can guide targeted training or process revisions for the involved cooks.

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Most popular questions from this chapter

The color of a person's eyes is determined by a single pair of genes. If they are both blue-eyed genes, then the person will have blue eyes; if they are both brown-eyed genes, then the person will. have brown eyes; and if one of them is a blue-eyed gene and the other a brown-eyed gene, then the person. will have brown eyes. (Because of the latter fact we say that the brown-eyed gene is dominant over the blue-eyed one.) A newborn child independently receives one eye gene from each of its parents and the gene it receives from a parent is equally likely to be either of the two eye genes of that parent. Suppose that Smith and both of his parents have brown eyes, but Smith's sister has blue eyes. (a) What is the probability that Smith possesses a blue-eyed gene? Suppose that Smith's wife has blue eyes. (b) What is the probability that their first child will have blue eyes? (c) If their first child has brown eyes, what is the probability that their next child will also have brown eyes?

\(A\) and \(B\) are involved in a duel. The rules of the duel are that they are to pick up their guns and shoot at each other simultaneously. If one or both are hit, then the duel is over. If both shots miss, then they repeat the process. Suppose that the results of the shots are independent and that each shot of \(A\) will hit \(B\) with probability \(p_{A}\), and each shot of \(B\) will hit \(A\) with probability \(p_{B}\). What is (a) the probability that \(A\) is not hit; (b) the probability that both duelists are hit; (c) the probability that the duel ends after the \(n\)th round of shots; (d) the conditional probability that the duel ends after the \(n\)th round of shots given that \(A\) is not hit; (e) the conditional probability that the duel ends after the \(n\)th round of shots given that both duelists are hit?

Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that (a) this student is female, given that the student is majoring in computer science; (b) this student is majoring in computer science, given that the student is female.

There are two local factories that produce radios. Each radio produced at factory \(A\) is defective with probability \(.05\), whereas each one produced at factory \(B\) is defective with probability.01. Suppose you purchase two radios that were produced at the same factory, which is equally likely to have been either factory \(A\) or factory \(B\). If the first radio that you check is defective, what is the conditional probability that the other one is also defective?

Consider two boxes, one containing 1 black and 1 white marble, the other 2 black and 1 white marble. A box is selected at random, and a marble is drawn at random from the selected box. What is the probability that the marble is black? What is the probability that the first box was the one selected, given that the marble is white?
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