# Probability of A given B

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:

1. Is there a word I can use instead of "given" to be clearer?
2. 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"?
3. Have people who attended high school in other countries had this difference made clear to them?
• 1. "knowing that" may be used instead of "given", I think. – Américo Tavares Aug 25 '10 at 12:06
• @Americo: I don't think that makes a difference. Knowing doesn't say how we know that occured – Casebash Aug 25 '10 at 12:40
• The first half-sentense in question 2. lacks a verb, I think. – Rasmus Aug 25 '10 at 13:30
• If a question with "given" isn't clear, there is always an option of using formal definition of conditional probability ($P(A|B):=P(A\cap B)/P(B)$). Anyway, I don't think there is any ambiguity in "given" — it's just that common-sense intuition fails here. – Grigory M Aug 25 '10 at 14:10
• I've seen "given that" replaced with "if" in those sorts of problem statements. – J. M. is a poor mathematician Aug 25 '10 at 15:47

## 4 Answers

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:

1. 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.

2. 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.

3. Not in the country I went to high school in :-)

• Re 2: Actually, that post seems to take "given" to mean "given" in the formal conditional probability sense. So, they only seem to object if a word other than "given" is used – Casebash Aug 25 '10 at 22:04

I'm hesitant to use an answer to respond, instead of a comment, because the answer seems to merely be a matter of semantics.

Let's look at a similar question:

"What are the odds that two people are of the same gender?"

And its counterpart:

"What are the odds that two people of the same gender, if we knew that one of them is a boy?"

The questions are inherently different. The answer to the first can be seen by charting all possibilities within the limitations (there are no limitations), and it can be seen that it's 2/4, or 50%

The second can be seen by charting all possibilities and eliminating the ones that aren't allowed (two girls), and we'll see that the answer is 1/3, or about 33%.

For a somewhat intuitive answer of "how" we know something, we can use a statistical model. Let's say we wanted to find the odds of the second question. We could conduct a survey, asking a perfect world: (where both genders are equally likely)

"Anybody who has two children, with one being a boy, answer if they have two boys or not."

This survey would give you the 33% that our chart-all-the-possibilities method would lead us to expect. There are no ambiguities because we see the answer very clearly real and evident.

But this is a maths site, and I brought up statistics, so I'm probably going to be killed in my sleep.

• If the question is taken to mean that, then it is clear that you will get that answer. However, my question relates to the question being ambiguous – Casebash Aug 25 '10 at 21:55

I don't think there is any ambiguity in the use of the word “given” here, but I do think that one needs to learn the way it is used in conditional probability.

If people are not familiar with how terms are used then of course there could be misunderstanding. There are some interesting comments about ambiguity in such questions in this wikipedia article.

(This sentence from the article made me smile: Many people, including professors of mathematics, argued strongly for both sides with a great deal of confidence, sometimes showing disdain for those who took the opposing view.)

Given is an accurate and acceptable way to phrase this question. The trick is to understand when conditions that are given affect the probabilities and when they do not. In this case, the probability is not affected. We know that the first person was born on a Tuesday. Now, in order to answer the question, "what is the probability that both people were born on the same day given that one was born on a Tuesday?", we only need to determine the probability that the second person was also born on a Tuesday. Therefore, the answer is 1/7.

Without the given condition, we arrive at the same answer. No matter what day of the week the first person's birthday is, there is still a probability of 1/7 that the second person shares that same weekday.

Finally, the answer would be quite different if phrased as "What is the probability that both people were born on a Tuesday?" Now the odds that the first person was born on a Tuesday is 1/7 (vs. 1/1 when "given" that the person was born on a Tuesday). The odds that the 2nd person was also born on a Tuesday is 1/7. So the odds that both were born on a Tuesday is 1/7 X 1/7 = 1/49