# Probability of familiarity with a popular item?

I want to know if there's a method for determining the likelihood that a person is familiar with the most popular item in a set, given the number of items in said set they are familiar with. Here's the best example I can think of:

Andrew walks into a room with 20 dogs. He's played with 3 of these dogs, but we don't know which dogs in advance. More people have played with Spike than any other dog in the room. Let's say that Andrew's circle of friends is 10 people (including himself), and 5 of them have played with Spike. What are the odds that Andrew has played with Spike?

I'm sure there's an existing formula for this type of problem, but there are so many probability formulae that I can't find the one I need.

-

Assume everyone, not just Andrew, played with 3 dogs chosen uniformly at random. Now we have no information to distinguish Andrew from any of the other 9 people who each played with 3 of the 20 dogs. So it seems reasonable to say the probability is 50% because 5 of 10 people have played with Spike and Andrew is one of them, i.e. Andrew may as well have been chosen at random. To break this down:

• Everyone randomly selects 3 of 20 dogs and plays with them.
• Andrew randomly selects 3 of 20 dogs and plays with them. (?)
• 5 out of the 10 people have played with Spike.
• No dog has been played with as many times as Spike. (?)
• The probability that Andrew has played with Spike is 50%.

On the other hand, if we don't know how many dogs anyone other than Andrew played with, or if we don't know by what distribution they were selected, and assuming Andrew's selection of dogs was uniformly random, then there is no way to use the information that 5 of 10 people have played with Spike. All that matters is that Andrew chose 3 dogs at random out of 20, so the probability that Andrew played with Spike is 15%. This reduces to:

• 5 out of the 10 people have played with Spike. (?)
• No dog has been played with as many times as Spike. (?)
• Andrew randomly selects 3 of 20 dogs and plays with them.
• The probability that Andrew has played with Spike is 15%.

I think a lot of other values are possible depending on what is assumed.

-
What I'm looking for with the popularity is the idea that if a certain item - in this case, a dog - is popular, people will talk about it and possibly share it, thus increasing the likelihood that Andrew will have played with the dog over other dogs. What if we assume that each of the five people talked about how much fun Spike was? – Fibericon Jan 13 '12 at 8:18
Suppose everyone plays with 3 dogs at random, but if they heard recommendations about a particular dog they with play with that dog too for a total of 4. Since Andrew played with 3 dogs he didn't hear any recommendations and chose them all at random, so the probability he played with Spike is 15%. – Dan Brumleve Jan 13 '12 at 8:35
On the other hand, if everyone who has heard a recommendation plays with 2 dogs at random plus a recommended dog, and everyone who hasn't plays with 3 dogs at random, then everyone has played with 3 dogs and we are back to the situation where there is nothing to distinguish Andrew, so the probability is 50%. – Dan Brumleve Jan 13 '12 at 8:41