B Decision for conditional probability instead of intersection of events

AI Thread Summary
The discussion centers on interpreting the phrase "Each microwave produced at factory A is defective with probability 0.05," highlighting the distinction between conditional probability and the intersection of events. It clarifies that if only factory A is considered, the probabilities of defectiveness align, but if multiple factories exist, the interpretation changes. The sentence lacks context to definitively indicate whether it refers to conditional probability or intersection, necessitating a broader understanding of the problem. The conversation emphasizes that conditional probability involves different probability spaces, which can lead to confusion among students. Ultimately, recognizing these nuances is crucial for accurately solving probability problems.
Peter_Newman
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Hello,

I have a question about the following sentence and would appreciate if someone could explain how to read out the conditional probability here.

"Each microwave produced at factory A is defective with probability 0.05".

I understand the sentence as the intersection ##P(Defect \cap Factory A)## rather than the Conditional Probability.

But for solving the problem, the Conditional Probability ##P(Defect|Factory A)## is needed.

Reading the sentence, what clue is there that it is a conditional probability and not an intersection?
 
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Peter_Newman said:
Hello,

I have a question about the following sentence and would appreciate if someone could explain how to read out the conditional probability here.

"Each microwave produced at factory A is defective with probability 0.05".

I understand the sentence as the intersection ##P(Defect \cap Factory A)## rather than the Conditional Probability.

But for solving the problem, the Conditional Probability ##P(Defect|Factory A)## is needed.

Reading the sentence, what clue is there that it is a conditional probability and not an intersection?
Not quite. This is actually quite subtle.

If we assume that there is only factory A under consideration, then we have:
$$P(Defect|A) = P(Defect \cap A) = 0.05$$That's because ##P(A) = 1##.

If, however, we assume there is also a factory ##B##, then
$$P(Defect|A) = 0.05$$But$$P(Defect \cap A) = P(Defect|A)P(A) \ne 0.05$$
 
I don't know that there is one particular way to recognize what they are asking, but to me it is pretty clear. Let me just write how I would say things:

##P(Defect|A)## "The probability that a microwave from factory A is defective"

##P(Defect \cap A)## "The probability that a microwave is both from factory A and is also defective"

##P(A|Defect)## "The probability that a defective microwave is from factory A"
 
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Peter_Newman said:
"Each microwave produced at factory A is defective with probability 0.05".

Peter_Newman said:
Reading the sentence, what clue is there that it is a conditional probability and not an intersection?

The sentence, in isolation, does not define a "probability space", so it does not reveal whether it refers to a conditional probability. Only the whole context of the problem would make that issue clear.

Considering that probability is a number that is assigned to a set of outcomes, the probabilities ##P(D \cap A)## and ##P(D | A)## both refer to assigning a probability to the set ##D \cap A##. The interpretations of the two probabilities differ with respect to the probability space under consideration.

For ##P(D \cap A)## we are considering some set ##S## of outcomes such that ##P(S) = 1## and where sets ##D## and ##A## are subsets of ##S##. (The set ##A## need not be all of ##S##. For example, as @PeroK says, there might be several factories that may produce defective items.)

For ##P(D |A)##, we are considering a probability space consisting only of outcomes in the set ##A##. In that assignment of probabilities, ##P(A) = 1##. But this is misleading notation since it suggests that there is only one function ##P## that assigns probabilities. It would be better to denote the function that assigns probabilities to the set ##S## as ##P_S## and the different function that assigns (nonzero) probabilities only to outcomes in the set ##A## as ##P_A##. So we have ##P_S(S)=1## and ##P_A(A) = 1 ##.

The formula ##P(D \cap A) = P(D | A) P(A)## is slightly misleading because the "##P##" appears to denote a single function. In terms of the sets ##S## and ##A## mentioned above, a better notation would be ##P_S(D \cap A) = P_A(D) P_S(A) ##

That notation makes it clear that conditional probability is a sophisticated concept that involves two different probability spaces. Many students make the mistake of thinking that a probability problem must involve only one function that assigns probability to a set of outcomes. By that way of thinking it is correct to refer to "the probability" of a set of outcomes because there is only one such probability. However, many problems involve several different probability spaces.
 
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