## Monday, March 06, 2006

### Is it worth the 25 cent toy?

Let's say there is a cereal that is putting out four different toys in their cereal box. Assuming the prizes are placed randomly and are evenly distributed, how many boxes of cereal would you need to buy on average (or, if you like, you should expect to buy how many boxes) before you got all four?

1. You have a 9% chance of getting all 4 on the first 4 boxes. How to translate that into number of boxes needed for 100% chance, i do not know right now. Brain fried today.

2. You should expect to buy 8 1/3 boxes.

You will find toy #1 in box 1. You then have a 3 in 4 chance of getting a new toy in box two. Which means you should expect to buy 4/3 boxes before you get a second toy.

Once you have two different toys, the chances of gettng a new toy is 50/50. So, you should expect to buy 2 boxes before getting a new toy.

The last toy has a 1 in 4 chance of showing up, so you should expect to buy four boxes.

That gives you 1 + 4/3 + 2 + 4 = 8 1/3 boxes.

3. I don't think any store would let you purchase 1/3 box. You'll need to but 9.

4. That's true, you would have to buy 9 ;-)

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