





  • $\begingroup$ 1. "knowing that" may be used instead of "given", I think. $\endgroup$ – Américo Tavares Aug 25 '10 at 12:06
  • $\begingroup$ @Americo: I don't think that makes a difference. Knowing doesn't say how we know that occured $\endgroup$ – Casebash Aug 25 '10 at 12:40
  • 2
    $\begingroup$ The first half-sentense in question 2. lacks a verb, I think. $\endgroup$ – Rasmus Aug 25 '10 at 13:30
  • 2
    $\begingroup$ 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. $\endgroup$ – Grigory M Aug 25 '10 at 14:10
  • $\begingroup$ I've seen "given that" replaced with "if" in those sorts of problem statements. $\endgroup$ – J. M. is a poor mathematician Aug 25 '10 at 15:47


So, specifically to answer your questions:




  • $\begingroup$ 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 $\endgroup$ – Casebash Aug 25 '10 at 22:04


Let's look at a similar question:


And its counterpart:








  • $\begingroup$ 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 $\endgroup$ – Casebash Aug 25 '10 at 21:55





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


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