Math Puzzle: The Cereal Box Surprise
Level of Difficulty: Undergraduate
Remember those special surprise toys you used to get as a kid inside the cereal boxes you bought? I used to spill the cereal out, retrieve the toy and put the cereal back in. No matter how cheap or crappy it was, I had to have it.
Now it’s time to revisit your childhood as a math-wielding adult. Suppose a box of cereal costs 5$, and each box has a toy in it. There are 5 different toys for you to collect; by collecting all of them you can assemble them together and create a giant robot. If the toys have equal probabilities of turning up - that is, each toy is 1/5 likely to appear in a randomly chosen cereal box - how much will you have to spend, on average, before you can assemble the giant robot of your dreams?
EDITED, 4/June/2009: a solution has been posted here. Hats off, and a giant robot, to our reader Andrew Brandon for his solution.
I think the answer is $57.083333…
I’ll keep my solution to myself for now so I don’t spoil the fun for other readers.
In that case, to keep the suspense running, I will refrain from commenting on your solution’s correctness for now.
For those who need a hint, try solving the following sub-problem first: if an experiment has N equally probable outcomes, how many times will you have to repeat the experiment (on average) to get a particular outcome, say outcome #1? For example, how many times must you toss a fair coin, on average, until you get Heads?
That’s how I did it.
I’m fresh out of an undergraduate probability course, so I’m enjoying these math problems; they’re keeping me in shape.