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## Random Numbers With Constraints

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #14822 Fun: (2.5) Difficulty: (2.18) Category: Probability Submitted By: j72883 Corrected By: Sunrose

One of ten different furbies was randomly placed into each box of some type of cereal. If a mother decided to buy this cereal for her children until she obtained at least one of each of the ten different furbies, what is the number of boxes of cereal that she is expected to purchase?

This is a problem that uses elementary probability. We know that the probability of selecting a different toy is 1/10. But, when we purchase that first box of cereal, we have a 100% chance of getting a toy that we don't have. Applying a probability distribution, we are expected to purchase 1/1 boxes of cereal initially. Next, we have a 90% chance of getting a toy that we don't have, so total we purchase 1/1 + 1/.9 so far. The probability of getting a toy that we don't have thus decreases with each successive purchase, down to a 10% chance of getting one that we don't have once we've gotten all the others. Thus, we are expected to purchase:
1/1 + 1/.9 + 1/.8 + 1/.7 + 1/.6 + 1/.5 + 1/.4 + 1/.3 + 1/.2 + 1/.1 boxes of cereal for a total of 29.29.

She will probably need to buy 30 boxes before she gets one of each toy.

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