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There are 16 distinct LEGO minifigures in a collectible series. They are sold individually in "mystery" bags so that you cannot tell which one is inside.

Minifigures

To get all 16 minifigures, what's the expected number of bag purchases I need to make?

For simplicity let's make the assumption that the probability of a single bag containing a specific minifigure always is 1/16.

In fact, you may sometimes be able to buy a case with 60 of these bags, guaranteed to contain the complete series. And you may swap figures on the Internet. Let's ignore such options for this question.

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  • Well you might need to add in how many you pick from a certain box, or is it a mix of 10 boxes, this all would increase the probabilities etc. I think using a bred new open box (which should contain min 2 full sets) you should be able to work out the math there I just feel the bags and 98% of the time get the one I want and if I have the time ill feel for a whole series in one hit from a Lego store and get a good 90% of the series (like the recent disney ones) – Adam Mann Pro Jun 29 '16 at 8:55
  • @Adam Mann Pro: Where I purchase these bags there are hundreds of them mixed in big boxes. And I don't have the talent to "feel" the contents with my fingers, hence the assumption that what I get is truly random. – Gruber Jun 29 '16 at 9:02
  • your odds seem quite high then, as they might be the duplicates from many boxes. if you were looking to sell them people are still willing to buy them opened (with all bags, accessories etc) as they will be guaranteed to get the one they want. – Adam Mann Pro Jun 29 '16 at 9:05
  • There are lots of folks on ebay who will sell you a complete unopened set. I bought one and it was complete and the bags were still perfectly sealed. – chicks Jun 29 '16 at 11:49
  • Even if you can feel out a couple of figures, that can increase your odds of getting a complete set quite considerably. For example, the wedge-shaped "skirt" of many female figures is often easy to identify. The short, non-jointed legs are only a little harder, as are accessories like 2x2 tiles. I recommend giving it another go: you might surprise yourself. – Kramii Jun 29 '16 at 20:53
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55 Bags

If the distribution is assumed to be completely random (not necessarily true), and of a large enough group to make your purchases not effect the total probability (likely) then this is an alternate of a classic math problem known as the Coupon Collector's Problem:

https://en.wikipedia.org/wiki/Coupon_collector%27s_problem

There are many discussions of this and other probability problems in the Mathematics Stack Exchange.

Avoiding all the maths The answer for 16 individual units is 55.

If they have a case of 60 with a guaranteed full set, personally I'd buy it rather than depend on the random distribution of bags.

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    This is a great explanation of the maths. Note, however, that the distribution of figures is not completely random: there are more of some figures than others. – Kramii Jun 29 '16 at 20:46
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    I agree it wouldn't be random, but it was an assumption listed in the question itself. Without knowing the actual distribution there is no way to solve the problem without making some assumptions. – Josh King Jun 29 '16 at 20:59
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16 Bags

I know the question was phrased as a sort of math problem, but most LEGO fans solve it as a sort of LEGO problem. The bags are opaque, so you cannot see what's in the bags, but almost everywhere they are sold you can feel the bags, and you can determine what figure is in the bag. With the exception of the recent Soccer Team figures, collectible minifigures are always unique in terms of what parts are in the bag. Just shake and squeeze the bag until the distinctive piece is found.

The problem with the math way: One can compute the probability of, say, picking two matching socks out of a drawer of miscellaneous socks, or of picking a whole collection of figures out of a box of figures. But there are a few complications:

  • Someone else has been there before you and squeezed the bags to find all the space marines.
  • Someone else has been there before you and grabbed a few at random, but then squeezed a few more bags to find the Princess.
  • The store takes what's left from the first box and jams it into the second box when the box runs low; or they don't even display them in the original box, making it impossible to know how many boxes are in play and how many figures have been picked.
  • There are usually 16 figures in a collection and 60 figures in a box; this means the figures cannot be evenly distributed. Some series have had as few as two of one figure in the box. Given that, it's entirely possible (though unlikely) to pick 58 figures and miss both of that elusive Unicorn Girl.

If squeezing the bags isn't an option, the only way to be sure is to buy a new, unopened box.

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