Solution to Puzzle No. 6 world cup medallions
For the question see "Puzzle No 6 - world cup medallions" in issue 6.
Firstly, we'll suppose that there are N different medallions to collect. This makes the working easier, plus the result we shall obtain is more general; we can simply put N= 22 at the end to obtain the answer to the problem.
As for the hint, we define the random variable Xn = Tn - Tn-1 where Tn is the number of medallions you have collected when you first own n different medallions.
The next step is to try to find the distribution of Xn i.e. to calculate [an error occurred while processing this directive]
for different values of n and j where [an error occurred while processing this directive]
Let us now fix such an n and j. In order that Xn = j we must have first picked j-1 medallions which we already had in our collection of n-1 different medallions, and then picked a new medallion. Assuming that each different type of medallion is equally likely to be picked at any stage (which we're told), then denoting p to be the probability that we pick a new medallion, and q to be the probability that we pick a medallion already in our collection, we have [an error occurred while processing this directive]
from which we see that [an error occurred while processing this directive]
Hence Xn has the geometric distribution with parameter p (defined as above). We may write this in symbols as [an error occurred while processing this directive]
It's not too hard to show that if a random variable [an error occurred while processing this directive]
then [an error occurred while processing this directive]
Hence [an error occurred while processing this directive]
The average number of medallions collected in total is therefore [an error occurred while processing this directive]
and, after a little rearranging, we obtain the result: [an error occurred while processing this directive]
So, putting N= 22 into this formula we see that on average the number of medallions we need to collect to obtain the full set is about 81.