Coupon collector's problem

The Collective Image problem, collectors issue, scrapbooks problem or even after the English name Coupon called Collector's problem deals with the question of how many randomly selected images of a collective image series are to buy to get a complete picture series.

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How many surprise pictures you have to buy the means to obtain a complete series of pictures when swap is excluded and each image in the goodie bag is equally likely?

The probability of obtaining a new image at the first random selection is the first case of the second random selection of a further image it is the third, the last time.

In order to obtain a second image that is new, you have to buy in the Middle Pictures, for the third and so on. Do I need an average of images, then:

Is the - th partial sum of the harmonic series.

For example, you have a six-sided dice throw an average of 14.7 times to get any eye number at least once, because with true

Packet

Collecting pictures are usually not individually but sold in small packages. In this case, it is guaranteed that within a packet occurs more than once a picture.

For example, if five images in one pack, the number of buying at an average of images is calculated as follows:

For the first image from the first packet, there are cheap from possible variations in order to obtain a unique image. The second image from the first packet now has (an image is already available) from cheap (no scene is repeated in packet ) possible variations to also be unique. For the first packet with 5 images this results in:

The second packet:

Etc.

General with album size and packet size:

Example

There were 150 different motifs on the Pokémon cards that you have to buy without knowing the motives ( the cards are individually wrapped). If we assume that the manufacturer of each motif is equally often packed, how many cards have the parents buy their child, on average, until it has all the motives?

Solution:

We first define the sample space:

Now we define random variables. This random variable is to assign each result, the number of purchases that must be made ​​to get a new, different from the previously purchased th card after th different card.

An example:

Then, since is purchased as the first card with number 11. The next of which various card bears the number 30 Then are four necessary purchases until a new (the third ) different map, in this case, the number 7, is acquired.

Now, the probability is determined that is, that the probability that the -th purchase a card is not yet present is acquired. Because we have no tickets available before the first purchase. Then is. Only the already acquired (first) card would be duplicated. If we carry this idea further, we eventually find in general:

This random variable is geometrically distributed. Then we have:

As expected value of we get:

If we define a new random variable

Indicating how many purchases must be made ​​to have all 150 cards. Since the expected values ​​of the individual, there exists also the expected value of the new random variable:

Therefore, we now know that parents have to buy 839 tickets in average until their child has all 150 cards.