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A breakfast cereal maker is putting toys in the boxes of cereals as part of a promotion. There are four different toys to collect. A child desperately wants to collect the whole set, and his/her parents are concerned about how many boxes of the same cereal they have to go through before he/she achieves this. It is assumed that each cereal box is equally probable to have any one of the toys inside, and that there is only one toy per box. On average, how many cereal boxes are needed to be bought so the child has the complete set?

Answer: We make use of the fact that the expected number of boxes needed to getting a different toy is 1/p, where p represents the probability that the current box bought is different to all previous ones.

So, for the first box purchased this is 1/1, since we know for sure this will not be a double.

The average number of boxes needed for the second (different) toy to appear is 1/(3/4) = 4/3 boxes.

For the third this is 1/(1/2) = 2 boxes.

For the final toy this is 1/(1/4) = 4 boxes.

Hence, the total expected number of boxes needed to be bought to have the complete set is simply the sum of all these, i.e. 25/3 boxes.

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