Question

In the 1980s, some boxes of Honeycomb cereal contained a digital watch. Suppose that
the probability that a box of Honeycomb cereal has a watch is 0.4, and that the presence
of a watch in a box is independent of one being in another box. If you purchase box
after box of Honeycomb until you have obtained two watches, what is the probability
that you purchase a total of 7 boxes of Honeycomb cereal?

Answer 1

In the 1980s, some boxes of honeycomb cereal contained a digital watch.

Suppose that a box of honeycomb cereal has a watch is 0.4.

The presence of a watch in one box is independent of being present in another box.

Now, one purchases box after box of honeycomb, until he or she obtains two watches.

To find the chance that total 7 boxes of cereal are purchased.

Now, the chance of not finding a watch in a box is 1-0.4, ie. 0.6.

Now, a total of 7 boxes are purchased, implies that the second watch is found in the 7th purchase only.

So, the first watch was found in any 1 of the first 6 purchases.

Now, we can choose the one purchase, out of the first 6, in which the first watch was found, in 6 ways. Now, for any choice, the first watch was found with chance 0.4. And the other 5 boxes had no watch, each with probability 0.6.

Then, the second watch is found in 7th box with chance 0.4.

So, by multiplication rule of probability, and by independence,

the chance that a total of 7 boxes were purchased, is

=

=

So, the probability that a total of 7 boxes were purchased, is 0.0746.