" The boxes of Crackle-and-Pop
Corn Flakes contain k different
prizes. How many boxes we have to purchase to have a 50/50 chance of getting
all the prizes?"
Let P[f; k, x] denote the probability that a sample of x boxes contains f different prizes. To estimate P[f; k,x] we simply need to repeat the experiment we carried out before, say T t imes, then count the number of times that exactly f prizes were found. That number divided by T is your estimate E[f;k,x] for the probability P[f; k, x] .
Below is an applet which will allow you to get an accurate estimate for this probability by allowing you to repeat the experiment thousands of times. The time it takes for the computer to do this is so small that you may even enter T=10000 if you wish. To help visualize your findings the applet will plot for you the estimate E[f;k,x] as a function of f, as f varies from 1 to k. Moreover, by clicking below any one of the vertical bars you will obtain the percentage of times the corresponding number of different prizes were found.
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To obtain a complete set of the prizes that are offered in a particular brand of Cereal you may have to purchase quite a few boxes. Using the following Applet you should be able to get an idea of just how many it takes. When you enter the number k of of different prizes an press Go it displays progressively increasing number of different prizes that it finds as it simulates the successive opening of the cereal boxes. After the display is completed, it states under it the number of boxes that it took to commplete the collection. You may also get the number collected after opening x boxes by clicking on the approximate position of x on the x axis or on any one of the blue dots of the display. |