(the) Road to the Deep East

AKB48 gives us an interesting mathematical question.

According to its official site, when a fan purchase one copy of their latest single 'Sakurano Hanabira-tachi 2008' at AKB48 Theater in Akihabara, one special poster is given to the person. There are 44 kinds of posters, and each poster has each picture of 44 members from AKB48.

You can NOT choose the poster when you purchase the copy, but when you collect all 44 kinds of posters, you will be invited to the special concert from this group, AKB48 Haru no saiten'(AKB48 Spring Festival).

http://ameblo.jp/akihabara48/entry-10075346906.html

This campaign is taken place from Feb 26 to 28 and March 1st to 2nd. When the poster runs out, this campaign ends.

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Some may say it's crazy, but as an EX-idol otaku, I can say it's a very common campaign to promote idol products. It brings me back to the sweet memories of a 15 year-old girl's hot pants 2 years ago... Now in this case, the special gift is a poster and special concert, not just a tiny card nor sticker without anything special like SweetS or HINOI team. AKB fans should think it's kinder than average campaign of minor idol groups from avex.

But not for AKB48 otakus, even for sane people like us, here comes a very intriguing question to think of. How many copies do AKB48 otakus have to buy to be invited to the Spring Festival?

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To set the things straight, let's start with a simple example.

Suppose that now this campaign was held for Perfume. Perfume has three members.

The conclusion first, the probability to collect all kinds of cards is given by the complement of the probability to get JUST one kind or JUST two kinds of posters.

For example, when you purchase 5 copies,
I.The possibility to get only one kind of posters
First, the possibility to get only a-chan's is :
(1/3)^5=0.0412
And Perfume has 3 members,
(1/27)*3=0.1235

II. The probability to get only two kinds of posters
the possibility to get a-chan's and kashiyuka's is:
(2/3)^5
And consider the all the combination, and exclude one kind only case.
(Kashiyuka + a-chan, a-chan+nocchi, nocchi+kashiyuka)
(2/3)^5* 3C2 –-01235; = 0.38272

III. The probability to complete all three kinds of posters(answer)
Given by the subtraction, 1 - (I+II).

1 - 0.38272- 0.1235 = 0.61728 =61.7%.

When the based number is small like Perfume's case, the probability to complete all is relatively high. Even five copies look enough to collect all posters.

However, now you have to complete 44.

To tell the answer first, when you buy only 44 copies spending 55,000 yens, the probability to complete all 44 kinds with 44 copies is...

44!/44^44

= 1 / 771,468,890,917,894,000

The number which we've never seen in my life appears here. It must be less than the possibility that Sasaki Nozomi-chan fells in love with me. The more the idol group expands, the more an otaku agonizes.

However, when you purchase more copies, this probability shoots up. The more copies you buy, the more the probability becomes.

For example, when you purchase 100 copies (125,000 yens): it rises until 0.4 %.
When you purchase 150 copies (187,500yens) : it rises until 22.1%.
When you purchase 200 copies (250,000 yens): it rises until 66.3%.
When you purchase 1,000 copies (1,250,000): it rises until 99.9%..

So, if you can spend over 1 million yens for this concert and purchase 1,000 copies, it must be sufficient to go to the AKB Spring Festival. When you can't afford it, 200 copies can be enough, but there's still possibility to miss the chance. If you buy 100 copies and are lucky enough, you may be able to the concert somehow. But it's difinitely impossible to go to the concert with only 44 copies 55,000 yens. The rich becomes the winner, and the winner takes all.

I have no idea how much money will be spent for this crazy campaign in Akihabara for real, but we should think we are lucky that we don't get involved into this otaku project.