A basic conditional probability issue: How do I know if I am dealing with P(A|B) OR P(B|A)

[Astronomer_X]

In example 3.2, we have the following question:

Suppose 1% of the UK population have a particular medical condition. Suppose also that 17% of the UK population are aged over 65. We know that the probability of someone diagnosed with this medical condition to be over 65 is 8%. What is the probability that someone over the age of 65 has this medical condition?

The events are defined as follows:

A: having the condition. P(A) = 0.01.

B: being over 65. P(B) = 0.17.

We are told that P(B|A) = 0.08

How are the last two statements of the questions not the same???

I am confused as to how the phrase ‘someone diagnosed with this medical condition to be over 65 is 8%’ means P(B|A) rather than P(A|B)? I thought it is describing the chances that given someone is diagnosed with the medical condition, they are over 65 - which is event A given B.

My reasoning is that event A is mentioned first. If that is not the case, I am confused by the intuition in recognising which event is first. I even thought that the final statement was P(B|A), because it asks the probability that someone over the age of 65 (event B) has the medical condition (event A).

 

[Steven G]

How are the last two statements of the questions not the same???

P(X|Y) is read as the probability of X happening GIVEN that Y has happened.

P(B|A) is the probability that someone is over 65 given that they have the disease. Imagine that you have ALL the people that have the disease in a room, and you find out that 8% of these these people are over 65%--that is exactly what P(B|A) means.

What is the probability that someone over the age of 65 has this medical condition?
Now imagine that you have everyone over 65 in a room. Now with this population you compute the probability of someone (in this group!) having the disease.

There is really a difference. In the 1st case your sample space is everyone who has the disease and you compute the probability that someone is over 65.
In the 2nd case your sample space is everyone over 65 and you want to compute the probability of someone having the disease.


Consider this scenario. You roll a die and you want the probability of getting a 6 given that you rolled an even number, P(6|even). This probability is 1/3 since you know that you rolled either 2, 4 or 6 (the even rolls).

Now consider P(rolled an even number| you rolled a 6). This probability equals 1, since if you rolled a 6, then you certainly rolled an even number.

P(A|B) and P(B|A) are not the same.

Did you ever hear of Bayes Theorem??

 

[Dr.Peterson]

I am confused as to how the phrase ‘someone diagnosed with this medical condition to be over 65 is 8%’ means P(B|A) rather than P(A|B)? I thought it is describing the chances that given someone is diagnosed with the medical condition, they are over 65 - which is event A given B.

My reasoning is that event A is mentioned first. If that is not the case, I am confused by the intuition in recognising which event is first. I even thought that the final statement was P(B|A), because it asks the probability that someone over the age of 65 (event B) has the medical condition (event A).

I would rephrase

the probability of someone diagnosed with this medical condition to be over 65​

as

the probability that a person diagnosed with the condition is over 65​

which means

P(over 65, given diagnosed) = P(over 65 | diagnosed) = P(B | A)​

What is "given" is the population in view, which is those who are diagnosed; that is A. And what is given comes second in the notation. I suspect that is your central difficulty: not understanding the notation P(B | A). If we wrote P(A | B), that would be the probability of A, given B.

It is not at all uncommon for the "given" (the condition) to be mentioned first in English; but it is always written second in the notation..

Often you need to rephrase (even in several steps) in order to make complicated English phrases clear enough to work with! Never assume that the order in English matches the order of the symbols. Recall, from algebra, "5 less than x" = x - 5.

 

[Astronomer_X]

How are the last two statements of the questions not the same???
Did you ever hear of Bayes Theorem??

Thank you, the line about distinguishing which are my two sample spaces is particularly helpful. And it's funny you say that because this was a practice question after I learnt Bayes theorum.