Don't shout it or they'll all want one
Zaphod is collecting figurines from cereal packets. Each packet of Space Crunchers contains one of 4 different models, each equally likely. Zaphod hates cereal, but loves himself, so keeps buying Space Crunch until he has all four models. What's the expected number of packets Zaphod consumes?
Give your answer to two decimal places.
The answer is 8.33.
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1 solution
For the first box, he will receive one of four, but it doesn't matter which one. Next, he will want one of the remaining three, then one of the remaining two, then he'll finally want the last one. So it will take him 4 / 4 + 4 / 3 + 4 / 2 + 4 / 1 times to get it. The reason that it's 4 divided by however many he wants is because that is the expected amount of times it will take to get that certain one. So the answer is 8.33.