Decision for conditional probability instead of intersection of events

In summary: In that case it is a mistake to refer to "the probability" o a set of outcomes. In that case we refer to "a probability" of a set of outcomes. In summary, The sentence "Each microwave produced at factory A is defective with probability 0.05" is a conditional probability because it is discussing the probability of a microwave being defective given that it is produced at factory A. This is different from the intersection of the event that a microwave is defective and the event that it is produced at factory A, as it is only looking at the subset of factory A. The key clue is the use of the conditional probability notation ##P(Defect|Factory A)##, which indicates that the probability is being
  • #1
Peter_Newman
155
11
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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  • #2
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$$P(Defect \cap A) = P(Defect|A)P(A) \ne 0.05$$
 
  • #3
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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  • #4
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.
 

Related to Decision for conditional probability instead of intersection of events

1. What is the difference between conditional probability and the intersection of events?

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. The intersection of events, on the other hand, refers to the occurrence of both events simultaneously. In other words, conditional probability takes into account the occurrence of one event, while the intersection of events considers the occurrence of both events together.

2. When should I use conditional probability instead of the intersection of events?

Conditional probability is used when we want to calculate the likelihood of an event occurring given that another event has already occurred. This is useful when we have additional information about the occurrence of an event, which can affect the probability of another event happening. The intersection of events is used when we want to calculate the probability of two events occurring simultaneously, without taking into account any additional information.

3. How do I calculate conditional probability?

Conditional probability is calculated by dividing the probability of the intersection of the two events by the probability of the given event. This can be represented mathematically as P(A|B) = P(A∩B)/P(B), where A and B are two events.

4. Can conditional probability be applied to more than two events?

Yes, conditional probability can be applied to more than two events. In this case, we would need to take into account the probability of the intersection of all the events involved. For example, if we have events A, B, and C, the conditional probability of A given that both B and C have occurred would be represented as P(A|B∩C) = P(A∩B∩C)/P(B∩C).

5. What are some real-world examples of conditional probability?

Conditional probability is used in various fields, such as medicine, finance, and weather forecasting. For example, in medicine, conditional probability is used to calculate the likelihood of a patient having a certain disease given their symptoms and medical history. In finance, it is used to assess the risk of an investment given the current market conditions. In weather forecasting, it is used to predict the likelihood of a certain weather event occurring given the current atmospheric conditions.

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