[Math] solve the following problem with the Law of Total Probability

probability

According to this link,

(…) Usually, in the elementary exercises in probability texts, there would be 2 such sub-experiments, one followed by the other. You would have to compute the probabilities in each sub-experiment and then put them together using the Law of Total Probability to get the answer. (…)

Is this correct as this doesn't seem to match the following problem:

Three machines A, B, and C produce respectively $50$%, $30$%, and $20$% of the total number of items in a factory. The percentages of the defective output of these machines are $3$%, $4$%, and $5$%. If an item is selected at random, find the probability that the item is defective.

To my understanding, this problem is a Total Probability Problem as :

  1. The sample space is partitioned into three sections which sum up to 100%.
  2. The event is spread across all of the three partitioned spaces.

But, what about the following problems?

  1. Three machines A, B, and C produce respectively $50\%$, $30\%$, and $10\%$ (the partitions sum up to 90%) of the total number of items in a factory. The percentages of the defective output of these machines are $3$%, $4$%, and $5$%. If an item is selected at random, find the probability that the item is defective.

  2. Three machines A, B, and C produce respectively $50\%$, $30\%$, and $20\%$ of the total number of items in a factory. The percentages of the defective output of 1st two machines are $3\%$ and $4\%$ (machine C is unknown). If an item is selected at random, find the probability that the item is defective.

Should and/or can I apply Law of Total Probability in these 3 problems? Why or why not?

Best Answer

Comment: Summing up what can and can't be solved, and common names of the relationships used. Please look in your textbook or class notes to find the names of the used for these relationships in your course.

"Three machines A, B, and C produce respectively 50%, 30%, and 20% of the total number of items in a factory. The percentages of the defective output of these machines are 3%, 4%, and 5%. If an item is selected at random, find the probability that the item is defective."

This problem can be solved. Writing intersections as products, we have:

$$P(D) = P(AD) + P(BD) + P(CD) = P(A)P(D|A)+P(B)P(D|B) + P(C)P(D|C) \\ = .5(.03) + .3(.04) +.2(.05) = 0.015 + 0.012 + 0.01 = 0.037.$$ Thus the overall defective rate of the factory is 3.7%. As you say, this is an application of the Law of Total Probability.

Often a follow-up question to this kind of problem would be as follows.

"The machines make harmless characteristic markings on the items they produce, so it is possible to tell which machine made each item. If an item chosen at random from the output of the factory is found to be defective, what is the probability it was made by machine A?"

We seek $P(A|D) = P(AD)/P(D) = 0.015/0.037 = 0.4054.$ That is, over 40% of the defective items produced by the factory are made by machine A. (Even though machine A has an error rate of 3% (the lowest of any of the machines), it makes a greater proportion of the items than do the other machines, so it makes over 40% of the defective items. This is a use of Bayes' Theorem.

A key relationship used throughout is sometimes called the General Multiplication Rule: $P(EF) = P(E)P(F|E).$ It is a direct result of the formula $P(F|E) = P(FE)/P(E),$ where $P(E) > 0.$ This is the Definition of Conditional Probability.

Problem (1) cannot be solved for the reason given by @lulu, and Problem (2) cannot be solved for the reason given by @joriki.

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