In high school, we were often given questions of the form: "What is the probability of A given B?" For example:
What is the probability that two
people were born on the same day given
that one was born on a Tuesday?
Intuitively, most people expect that knowing someone was born on a Tuesday doesn't give you any information, as it is no different from being born on a Monday or Wednesday. However, the answer given is that we are looking at pairs of people with at least one born on Tuesday (13 out of 49) and only one of these pairs has both people born on the same day, so the answer is 1 in 13. If by "given one was born on Tuesday", you mean that we the birthdays of the two people are uniformly distributed between all possible pairs where at least one was born on a Tuesday", then this analysis is correct.
However, the question stated just says that we are "given" this fact. It doesn't explain how we came to know this fact. Lets suppose someone blurted this fact out randomly. Generally it wouldn't be because they looked at groups of people and discarded them until they found a pair where at least one was born on a Tuesday. Instead, I think it would be better to model it as someone randomly getting a pair of people, then making a true statement about them. For simplicity, we will assume they statements are of the form: "One of these people was born on a (INSERT DAY)". We will assume that if they were born on different days, each is equally likely. In this model, the statement that someone was born on a Tuesday actually makes no difference to the chance they were born on the same day.
It seems that part of the problem comes from giving the word "given" (often represented by the symbol |) a formal definition. People expect it to have the same meaning as the English word given. Anyway:
- Is there a word I can use instead of "given" to be clearer?
- Do mathematician's object to the problem being stated this way, or is this considered to be clear? If they do, why is the symbol "|" often pronounced as "given"?
- Have people who attended high school in other countries had this difference made clear to them?
Best Answer
It is very true that statements and problems on conditional probability are often presented in an ambiguous way. The problem isn't specifically with the term "given", I believe, but rather with the fact that the presentation does not make it clear what the sample space is and what are the distributions (typically some things are assumed, without comment, to be uniform).
So, specifically to answer your questions:
No, not that I'm aware of. If the variables are clearly defined, the use of "given" to indicate conditional probability is common and perfectly fine.
If I understand the question correctly (there seems to be a word missing), then most mathematicians do not regard such phrasings as you have mentioned to be sufficiently clear. See Peter Winkler's comments on exactly this kind of problems.
Not in the country I went to high school in :-)