Math is math and it is supposed to guide real-world decisions......but not always. There are times one would rather stick to the randomness like "what if it was me..." fantasies, even the math and probability have already given enough hints to stop.

Generally, this blog is supposed to be a reminder using freshman-level math (calculus), because I constantly forget the lesson of math when I encounter real-world issues, as in the case of "Nama Jashin" collection. Here I provide this blog simply to

• remind myself not to spend too much money on it.
• to make it clear to myself that how the randomness of "Nama Jashin" collection is leveraged to sale promotion, which is arguably the oldest, but most effective trick for commercial promotion sale.

More specifically, main content of this blog is as follows

• to provide an expectation estimation of nama jashin collection in a basic problem setting. Answers the question of "How many CDs I need to buy on average to get the desired distinct type nama jashin ?"
• to provide probabilistic bound of a instance case, in other words, to determine how lucky or how unlucky you are, given a result of your collection. Answers the question of "How many CDs you needs to buy if you want to get the desired one nama jashin at a high probability, say 99%"
• to provide also, estimation in some more complicated settings and how to deal with new settings. Answers the question of "How many CDs you need to buy on average to collect multiple distinct types?"

Outline of content

Intro and background

"Nama Jashin" (生写真, or なまじゃしん in hiragana) is a Japanese word, and could be roughly translated as "not-photoshoped raw photos", or "photos that has not been put into commercial usages before". It is commonly seen as a sale promotion scheme to put nama jashin into musical CDs or other commercial products. Usually, Japanese musical singers and bands will put nama jashin of themselves into the limited edition of their album CDs. And if one wants to collect a full set of nama jashin, one simply has to buy more CDs.

A full set Nama Jashin (three types in total), of 白石麻衣 (Shiraishi Mai) from Nogizaka46. (photos from Internet)

Note that, the Japan music market is strangely special and prosperous, so is the strangely special promotion scheme. Usually, for an album or a single, there would be multiple versions of it, limited edition, normal edition, with CD, or with CD and DVD. For limited editions there might also be type A, type B... or even type C and type D. Taking an extreme case, the recently-debut musical group Yoshimoto-zaka 46, consisting of former Owarai talk-show comedians, has in total 46 types of their first single CD.

Now you might wonder, what is special about the limited editions? Well, one would expect to see 特典(extra content, or special gift) in limited editions, one of such extra content is the nama jashin. Generally speaking, one piece of nama jashin is put into the limited edition. But the simple sale promotion trick here is that, there could be multiple types of nama jashin, and the record company encourages you to buy multiple limited edition CDs to collect all kinds of nama jashin .

I will talk about the randomness on how one is supposed to get the full set of nama jashin, the expectations, and the probabilistic bound. I will start with a simple setting and then move to advanced "more real-world" cases.