When I was an undergraduate, I was walking through the town centre one morning when a journalist from the student newspaper asked my views on the Rag Mag that had just appeared. When the report came out, six students' views were quoted. Three of the six happened to be from the same cohort of six maths undergraduates at my college. My Director of Studies concluded (jokingly) that this disproportionate representation of his tutees showed that we were spending too much time out and about when we should have been studying.
Here is a similar scenario, based on a recent experience. Suppose that I enter a lottery. It is open to, say, 10,000 people, but I do not know how many choose to take part. As it happens, I win the lottery. It seems to me that I can conclude that (with a high degree of probability) only a small proportion of those who were eligible chose to enter the lottery. If all 10,000 people entered then my chance of winning was only ,00001, whereas if 5 people entered it was 0.02. Since I did win, my estimate of the likely number of entrants will be at the lower end.
Now, you are my friend and you are equally friendly with all 10,000 potential participants. I tell you about it, I think you should agree that I am right in my conclusion. But you cannot come to the same conclusion: whichever of the 10,000 people had won the lottery would have told you of their success, so you can't deduce anything from what I have said to you about the likely number of participants.
So - I have good reason to come to a conclusion (with a high degree of probability). You agree that I am right to come to this conclusion, but also you have no reason to suppose that the conclusion is valid. How can this be?
Apologies if I am missing something but I think this analysis is sound. Have I read about this somewhere?
However, I haven't yet managed to persuaded anyone else to agree with me (which is perhaps ironic, or perhaps I am just wrong). Comments welcome!
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