Collecting Toys From Cereal Boxes

Suppose a manufacturer of breakfast cereals uses a promotion to increase sales. The promotion involves randomly p;acing one of four models of toys in each box of cereals. How many boxes would a customer need to buy to collect all four toy?
We use the geometric distribution to model the process of collecting the toys.
The first box of cereals is guaranteed to contain a toy you do not have, and this starts your collection.
The second box contains one of four toys, three of which the customer does not have. The probability of the box containing a different toy is  
\[\frac{3}{4}\]
  so we can use the geometric distribution  
\[Geo (\frac{3}{4})\]
  and the expected number of boxes the customer needs to buy to collect a different toy is  
\[\frac{1}{3/4}= \frac{4}{3}\]
.
The next box contains one of four toys, two of which the customer does not have. The probability of the box containing a different toy is  
\[\frac{2}{4}\]
  so we can use the geometric distribution  
\[Geo (\frac{2}{4})\]
  and the expected number of boxes the customer needs to buy to collect a different toy is  
\[\frac{1}{2/4}= 2\]
.
The next box contains one of ffour toys, one of which the customer does not have. The probability of the box containing this toy is  
\[\frac{1}{4}\]
  so we can use the geometric distribution  
\[Geo (\frac{1}{4})\]
  and the expected number of boxes the customer needs to buy to collect the last toy is  
\[\frac{1}{1/4}= 4\]
.
The total number of boxes the customer would expect to have to buy is 1+4/3+2+2+4=25/3.

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