Expected number of trials before all cards are collected

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In summary, the cereal company is giving out five baseball cards with equal probability in every box of cereal. The expected number of boxes you have to buy before you collect all five cards is 137.
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xcrunner448
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Homework Statement



A cereal box company started a promotion in which they put one of five different baseball cards in each box of cereal. All cards occur with equal probability. What is the expected number of cereal boxes you have to buy before you collect all five cards?

The Attempt at a Solution



The expected value is the sum from k=5 to infinity of k*(probability of taking exactly k boxes to collect all 5 cards). However, I cannot find a general expression that gives that probability. For k=5, it would be (0.2^5)*5! = 0.0384, because the probability of a certain card in each box is 0.2, and there are 5!=120 different ways to collect all five cards. I cannot figure out how to calculate this probability for k>5. I have tried the same method as I used for k=5, that is, (0.2^k) times the number of ways of collecting all five cards, but because some cards must be repeated 1 or more times I am having difficulty finding out how to calculate that.
 
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welcome to pf!

hi xcrunner448! welcome to pf! :smile:

hint: what is P(not ending at k) ? :wink:
 
  • #3
Hi tiny-tim, thanks for your response! I also had that idea, but I could not figure out a general expression for that either. For k=5, the probability of not getting, say, card #1 by the fifth box is (0.8)^5. I thought I'd multiply that by 5 to cover the all 5 possible cards, but that adds to greater than 1. I realized that I would then have to subtract the overlap (the probability of not getting cards #1 and 2, or the probability of not getting cards #2,3, and 4, and so on). But that got very complicated very quickly. There may be a much better way of finding it, but I could not think of it.

In the meantime, however, I found "[URL [Broken],[/URL] which describes the problem perfectly. It is an interesting derivation, based on the expected number of boxes to find the ith card after finding (i-1) cards. It gave me the answer of 11.416666=137/12, which I believe is correct.
 
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1. How is the expected number of trials calculated for collecting all cards?

The expected number of trials to collect all cards is calculated by multiplying the number of cards by the natural logarithm of the number of cards, which is also known as the Coupon Collector's Problem formula. This assumes that each card has an equal chance of being collected in each trial and that the trials are independent.

2. Is the expected number of trials the same for any number of cards?

No, the expected number of trials will vary based on the number of cards. As the number of cards increases, the expected number of trials also increases. For example, if there are only 10 cards, the expected number of trials would be much lower compared to if there were 100 cards.

3. Can the expected number of trials be used to predict when all cards will be collected?

No, the expected number of trials is an average and does not guarantee that all cards will be collected within that number of trials. It is possible for all cards to be collected before or after the expected number of trials.

4. How does the probability of collecting all cards change as the number of trials increases?

The probability of collecting all cards increases as the number of trials increases. This is because with each trial, there is a chance that a new card will be collected, thus increasing the overall probability of collecting all cards.

5. Can the expected number of trials be applied to other scenarios besides collecting cards?

Yes, the concept of expected number of trials can be applied to other scenarios where there is a fixed number of objects to be collected or observed. For example, it can be used to calculate the expected number of coin flips before getting a specific sequence or the expected number of dice rolls before getting all numbers from 1 to 6.

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