Each "purchase" consists of selecting an object out of the bin at random and then replacing it.
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X X X is the discrete random variable that represents the number of purchases until each of the n n n objects is selected at least once. Typically, the "bin" is interpreted to be some pack in which the contents are randomized. One of the key aspects of this problem is that the selections are mutually independent selection with replacement. Practically speaking, this means that objects can be selected more than once.
However, in this problem, multiple selections of the same object do not matter; only selections of distinct objects matter. The linearity of expectation page contains several problems relating to the coupon collector problem. These problems should be attempted before moving on to the generalization of the coupon collector problem, below. For the coupon collector problem as stated above, the expected value of the number of purchases required in order to select each of the n n n objects at least once is:.
Show Proof Let X X X be the discrete random variable that represents the number of purchases until each of the n n n objects is selected at least once. Then by probability by outcomes ,. For the coupon collector problem as stated above, the expected value of the number of purchases required in order to select each of the n n n objects at least once is approximated as:.
This approximation tends to be very accurate, and is more accurate with larger n n n. The collectible card game Arcane: The Congregation just came out with a new set of cards. Cards are sold in sealed packs, and each pack contains a single random rare card. The contents of each pack is independent of every other pack, and each rare is equally likely.
Coupon Collector’s Problem
If there are 55 distinct rare cards in the newest set, what is the expected number of packs needed to obtain at least one copy of each rare card? Round your answer to the nearest integer. This problem, as described, fits the format of a coupon collector problem. Packs are independent, the contents are random, and each rare is equally likely.
Fortunately, the problem as stated only requires an answer to the nearest integer. Therefore, the approximation can be used:. If a pack contains a mythic card, each of the 15 mythic cards is equally likely. If each pack of Arcane: The Congregation is independent of all other packs, what is the expected value of the number of packs needed to obtain at least one copy of each mythic card?
Note : The coupon collector problem page contains an approximation that makes the solution easier to compute, and the approximation is sufficient to obtain the correct answer. Submit your answer. Jorge has an N N N -sided fair die and wonders how many times he would need to roll it until he has rolled all the numbers from 1 1 1 to N N N in any order. Citing Literature. Volume 15 , Issue 2 November Pages Related Information.
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Question in probability (coupon collector's problem)
Browse All Figures Return to Figure. Previous Figure Next Figure. Email or Customer ID. Forgot password? Old Password. New Password. There's a discussion going on about this in relation to Party's latest promotion and maybe somebody here can help us discussion starts at this post, but the posts above may also be relevant. Thanks - Juk. Originally Posted by jukofyork. Last edited by BruceZ; at AM.
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Reason: Added R script, data, and latex formula. There's a discussion going on about this in relation to Party's latest promotion and maybe somebody here can help us discussion starts at. I added the R script and the output data to my original post. It should agree with your data, and it looks like it does. What tool did you use to compute these? Is there really no reasonable way to calculate this just using counting?
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Suppose that's the reason for using inclusion-exclusion instead. Thanks guys!
Coupon Collector Problem
Originally Posted by BruceZ. I'm guessing this is a continuous approximation - where does this come from? Originally Posted by sallymustang. I'd tweet you some photos, but I might want to run for congress someday. BB code is On. Smilies are On. Forum Rules. All times are GMT