## Solution to Puzzle No. 6 world cup medallions

Submitted by plusadmin on January 1, 1999For 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* *X _{n}* =

*T*-

_{n}*T*

_{n-1}where

*T*is the number of medallions you have collected when you

_{n}*first*own

*n*

*different*medallions.

The next step is to try to find the *distribution* of *X _{n}* 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 *X _{n}* =

*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 *X _{n}* 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.