So there are no stores that sell CMFs near to you, and you want to get a complete set. So, you go online to buy CMF packets, and buying random packets is often cheaper than buying identified minifigs. You know that if you buy 16 packets, it's very unlikely that you'll get all 16 figs (actually your chances would be 1 in 20,922,789,888,000). So, you wonder, how many packets should you buy?

Back then when series 6 was just new, Weil, Vexorian, and I discussed on the CMF series 6 thread about the probability of getting a complete set of 16 figs out of n completely random packs. Our problem is a specific case of the coupon's collector problem, and here is a blog post elsewhere about our "Minifigure Collector's Problem". However, this blog post didn't help us find the probability of getting a complete set after buying n packets, so we got to work. We all got consistent results, but took different approaches that have different applications. However, I just thought that it would be convenient to have all this info summarized in one post, so people interested in it don't have to dig in the series 6 thread whenever a new series arrives.

I will start with my approach, since it's the one with more limitations in our specific case. Assuming that all the figs come in equal distribution (which is not the case), the probability of getting all the figs out of n random packets is given by this formula:

%29*%2816-i%29^n*%28-1%29^%28i%2B1%29]]%2F%2816^n%29"]

In case you are interested in the Math behind it, I calculated this formula manually using the inclusion-exclusion principle.

Go ahead, try some values of n! You'll notice that the probability is obviously 0 for n in {1,2,3,...,15}, extremely low for n=16, and that the least amount of packets that you would need to buy to get a 50% chance would be 51.

Vexorian, on the other hand, used dynamic programming to consider the actual distribution weights of the figs. Although the probabilities vary depending on the distribution (more even distributions tend to make the probability higher), and this means that the probabilities might vary from series to series, the distribution for series 7 is the same as it was in series 6 (3 figs come 5 times in a box, 6 figs come 4 times in a box, and 7 figs come 3 times in a box), and I think that we can expect this distribution to continue to be used, so Vexorian's calculations are still the same for series 7 (and any series that has the same distribution scheme, for that matter). He made a table with the probability calculated for different values of n, and a blog post in which he further discusses the work he made. I'd summarize the specifics, but I don't know that much about programming, and you're better off if you read about his own work written by himself.

Vexorian's program is also useful to determine the expected number of bags you would need to buy to get any specific arrangement of figs (3 Galaxy Patrols, 5 Bunny Suit guys, and 2 Daredevils, for instance), so it has wider applications.

Finally, Weil also wrote a program to calculate the probabilities, which, however, uses a completey different approach than Vexorian's. He used the Monte Carlo Method. He even made a graph of the probability to get a complete set out of n packets, comparing our different approaches and showing that they are consistent. Take a look if you'd like to see how the probability changes as n grows.

Weil's Graph

So, what do we all conclude?

As you can see, you'd need to buy a fairly high amount of figs to get a good chance to get 16 minifigs. We found it interesting and fun to calculate said probabilities, but our results show that you're better off going bag feeling or buying figs off Bricklink. Buying random figs is not economically practical.

One important note, though. When you order random packets online, the figs you get are probably not that random. They probably come from the same box or a group of boxes, and all boxes have all the minifigs, so the selection that you get is not 100% random. However, it could also be the case that the seller tampers with the distribution to lower your chances of getting a complete set. It goes both ways.

Still, these numbers give you a guide, and it goes to show that you're better off identifying bags or buying lose figs. Good luck in your figure hunting!

Edited June 5, 201212 yr by johnnyvgoode

Also, if you order random packets online it's very likely that the seller already picked out the more desirable ones to sell at a higher price. Well, basically what you said, but I think it's more common for this to be the case than not.

Edited June 5, 201212 yr by sharky

Also, if you order random packets online it's very likely that the seller already picked out the more desirable ones to sell at a higher price. Well, basically what you said, but I think it's more common for this to be the case than not.

Yes, that is right, although less likely depending on the seller. For instance, if you order online from Lego, or from TRU.

A couple of weeks ago I collected the final Series 3 figure - the indian chief - I had bought them all off E-Bay as random packets, for no more than $4 each, sometimes it was 3, I was actually really surprised (and really bloody happy) That I managed to get 14 out of 16 as mystery bags and only needed to order the Mummy and Indian separately - and for cheaper than an individual mystery bag, so Series 3, a year after it's release, I was able to collect all of them with only about 12/13 doubles - (1 extra fisherman, 4 samurais (so not complaining about that), an extra snowboarder, an extra mummy (I bought 2 when I bought the first so it's not really counted), 3 extra Elves, 1 extra Tennis Player, an extra Dancing girl/Islander girl and finally, one extra Rapper... So in hind sight, series 3 = really lucky for me!

As for the rest, haha! Well, I managed to get Series 4 - but had to order the Ice Skater online and also got about 23 doubles lol

And to be honest, the rest I got tired of getting so many spares that I've started buying them on ebay identified - but still will buy a few mystery ones when I'm doing a Lego haul lol - The whole point of it is the fun of not knowing what we're getting so I still like to get at least 3 every week.

As for series 1 and 2, well, I have to get them identified, but I highly doubt I'll be able to afford one of the zombies, bugger!

Anyways, all your mathematics hurt my head lol But I'm 70% stupid.

You know that if you buy 16 packets, it's very unlikely that you'll get all 16 figs (actually your chances would be 1 in 20,922,789,888,000). So, you wonder, how many packets should you buy?

Actually, not.

1/16! is the chance you might have if you take from a market with at least 16x of every minifig, assuming they are fishing out from exactly 16*16=256 minifigures, which means at least 5 boxes of minifigures. That number doesn't apply.

If you want to know exactly how many minifigures you get, you must rely to the boxes infos... so let's say K = 3.75 (60/16) minifig of each type per box.

The first one is 100%. The second pick is 15K/59 which means 95.34% of chances to get two different minifigs.

The third one is .9534 *14K/58.

Counting on , this is the chance you have assuming the online shop picks 16 random minifigures from the same box.

1 = 100%

2 = 95.34%

3 = 86.3%

4 = 73.81%

5 = 59.31%

6 = 44.48%

7 = 30.89%

8 = 20.05%

9 = 11.57%

10 = 5.96%

11 = 2.68%

12 = 1.03%

13 = 0.32%

14 = 0.08%

15 = 0.013%

16 = 0.001%

So actually you got only about 1 / 100.000... which still doesn't helps since you can't buy 100.000 minifigures on line, but with this table you can guess how many minifig you may want to buy knowing the chances to get one double.

Actually, not.

1/16! is the chance you might have if you take from a market with at least 16x of every minifig, assuming they are fishing out from exactly 16*16=256 minifigures, which means at least 5 boxes of minifigures. That number doesn't apply.

If you want to know exactly how many minifigures you get, you must rely to the boxes infos... so let's say K = 3.75 (60/16) minifig of each type per box.

The first one is 100%. The second pick is 15K/59 which means 95.34% of chances to get two different minifigs.

The third one is .9534 *14K/58.

Counting on , this is the chance you have assuming the online shop picks 16 random minifigures from the same box.

1 = 100%

2 = 95.34%

3 = 86.3%

4 = 73.81%

5 = 59.31%

6 = 44.48%

7 = 30.89%

8 = 20.05%

9 = 11.57%

10 = 5.96%

11 = 2.68%

12 = 1.03%

13 = 0.32%

14 = 0.08%

15 = 0.013%

16 = 0.001%

So actually you got only about 1 / 100.000... which still doesn't helps since you can't buy 100.000 minifigures on line, but with this table you can guess how many minifig you may want to buy knowing the chances to get one double.

My head hurts :s lolol

Actually, not.

1/16! is the chance you might have if you take from a market with at least 16x of every minifig, assuming they are fishing out from exactly 16*16=256 minifigures, which means at least 5 boxes of minifigures. That number doesn't apply.

If you want to know exactly how many minifigures you get, you must rely to the boxes infos... so let's say K = 3.75 (60/16) minifig of each type per box.

The first one is 100%. The second pick is 15K/59 which means 95.34% of chances to get two different minifigs.

The third one is .9534 *14K/58.

Counting on , this is the chance you have assuming the online shop picks 16 random minifigures from the same box.

1 = 100%

2 = 95.34%

3 = 86.3%

4 = 73.81%

5 = 59.31%

6 = 44.48%

7 = 30.89%

8 = 20.05%

9 = 11.57%

10 = 5.96%

11 = 2.68%

12 = 1.03%

13 = 0.32%

14 = 0.08%

15 = 0.013%

16 = 0.001%

So actually you got only about 1 / 100.000... which still doesn't helps since you can't buy 100.000 minifigures on line, but with this table you can guess how many minifig you may want to buy knowing the chances to get one double.

Yes Itaria, as I indicated, these calculations don't consider the distribution to come from the same box, but rather completely random. This applies more to the scenario in which you just buy random figs from different places hoping to complete a set, though, but it gives you an estimate. However, when you order online, you never know if all the figs come from the same box.

And yes, the chance I put there is based on the equal distribution figs, but it gives you a good estimate. I don't know the precise number for the actual weighed distributions, but Weil and Vexorian did that work, and that's why I linked to it.

Edited June 7, 201212 yr by johnnyvgoode

Hmm... I guess I would add in one other variable - returns.

For example, I've read reports of people buying whole boxes of CMF from stores only to feel thru them at home and ensure a complete set. Then they just return the extras. One could do the same thing with an online order, since you're not opening the product, just feeling thru the bags. Major retailers who probably don't filter thru the CMF for the rare stuff probably also don't have a strict return policy.

I know the point of the thread was really more of a math question, but variables exist when collecting a full set of CMF. The idea that they're random only really applies in certain instances.

I know the point of the thread was really more of a math question, but variables exist when collecting a full set of CMF. The idea that they're random only really applies in certain instances.

Precisely.

This is more of a guide for people who want to know about how many random packets they should buy, but as we have mentioned, there are many instances in which the packets will not be completely random.

Precisely.

This is more of a guide for people who want to know about how many random packets they should buy, but as we have mentioned, there are many instances in which the packets will not be completely random.

Aside from opening a sealed box, I don't think there are any ways for the CMF figures to be truely random then.

If you buy from a store, they could have been felt thru already. If you buy online, they could have been felt thru already. So, really the absolute best way to know for sure that they're random in the first place is to find a new, sealed box - open that and you have the most accurate gauge. However, at that point, you might as well just feel thru the bags for a complete set.

Aside from opening a sealed box, I don't think there are any ways for the CMF figures to be truely random then.

If you buy from a store, they could have been felt thru already. If you buy online, they could have been felt thru already. So, really the absolute best way to know for sure that they're random in the first place is to find a new, sealed box - open that and you have the most accurate gauge. However, at that point, you might as well just feel thru the bags for a complete set.

There are people who just buy X number of packets online and try to get a complete set. If I remember correctly, some guy talked at some fan convention about buying a huge number of bags (about 120?) from series 2, and complaining that he got no Spartans! However, there are some important points to make here. For instance, take S@H, where some time ago you had a top limit of 16 packets. Many people who expected to get a complete set only by buying 16 didn't know the low chances that they were taking. The bags are not that random, but at the same time, they could have been tampered with to take a specific fig out of them, as we have mentioned. So, you conclude the same point that we did. It's best to go bag feeling, eBaying, or Bricklinking.

Edited June 12, 201212 yr by johnnyvgoode

There are people who just buy X number of packets online and try to get a complete set. If I remember correctly, some guy talked at some fan convention about buying a huge number of bags (about 120?) from series 3, and complaining that he got no Spartans! However, there are some important points to make here.

... such as the fact Series 3 didn't have a Spartan in it. No wonder that guy was so unlucky...

There are people who just buy X number of packets online and try to get a complete set.

Yes but I believe hawkman when he says that when you buy online bags may have been felt.

I will discover it right now. I am buying with friends 35 S3 packs for as much as 1.69€ from a german site.

If I find no elves, well, I will get my reply.

... such as the fact Series 3 didn't have a Spartan in it. No wonder that guy was so unlucky...

Right, he was talking about series 2 in a discussion about the lack of codes of series 3. Correcting it right away.

Edited June 12, 201212 yr by johnnyvgoode

Are they still arranged in the box the same way as previously ie all the same figure sequentially.

I am a feely guy now and so only get around four or five repeats.

A few series ago I discovered that if you find an unsealed box which is common here in Target stores

then open it and take the first one from one side, then move forward three and take another and forward three and so on...

I always ended up with the bulk selection and on my next visit did feely to get the rest.

Are they still arranged in the box the same way as previously ie all the same figure sequentially.

This is something that I have always wondered. So which series were arranged in that way, as far as you know?

I haven't really been trying to pick up a full series, but since I cherry pick individual types, I have only been wrong once.

To add to the original post, I noticed that the shops in my area typically sell figures from 1 box from behind the counter, so the total amount of figures is fixed and no one has gone through them to pick the rare ones. This is not covered by the original post, so I took a few minutes to simulate what happens in that case.

The simulation consists of filling a box with the known distribution of figures (I took the series 6 distribution mentioned by WhiteFang in his review) and then counting how many you have to open before you have a complete set. This I repeat 100.000 times and the probabilities are directly calculated from that. The graph is below, and the interesting observation is that you still require quite a lot of packets (35-36) to have a 50% chance of getting a complete set, but quite a bit less than in the 'open' situation. You're pretty much guaranteed to get to 100% before you reach 60 packets, simply because you bought the entire box in that case :).

If I want an entire series, then I just buy an entire box. Good luck!

I think that the only flaw with these views are the feel method. I use the feel method on my figures, and have only been wrong twice. Once with the Hazmat Guy (thought it was a Hockey player), and once with the Ocean King (I thought the blunt end of the trident was the blunt end of the spear, plus the trident felt longer than the one that I found before.

To add to the original post, I noticed that the shops in my area typically sell figures from 1 box from behind the counter, so the total amount of figures is fixed and no one has gone through them to pick the rare ones. This is not covered by the original post, so I took a few minutes to simulate what happens in that case.

Thanks for adding to the work we've done.

I think that the only flaw with these views are the feel method.

Again, we're speaking about a 100% random choice of figs, as opposed to using the feeling method to find the figs.

It's great to see that you took the time to calculate all this.

Again, I think it's fine that you took the time to calculate this, but it's not a very good "real world" example. Unless you see that store employee open the sealed box right in front of you, it's scienticially inaccurate to assume the box has not been picked thru. If there's even the slightest chance that it has been, then the data is contaminated.

I have to admit, this really feels like more of a math exercise than a Lego product exercise, as poeple buying packs one-by-one to get a complete set from a freshly opened box is not at all a common practice.

On 6/30/2012 at 1:13 PM, Hawkman said:

Again, I think it's fine that you took the time to calculate this, but it's not a very good "real world" example. Unless you see that store employee open the sealed box right in front of you, it's scienticially inaccurate to assume the box has not been picked thru. If there's even the slightest chance that it has been, then the data is contaminated.

I have to admit, this really feels like more of a math exercise than a Lego product exercise, as poeple buying packs one-by-one to get a complete set from a freshly opened box is not at all a common practice.

Yup, as I said, the approaches have limited applications and do not refer to the probability of getting a complete set under any kind of circumstances, but only to the probability of getting all out of a completely random selection. In fact, if you buy online, the boxes have not been picked through by people looking for a specific figure. Trying to complete a set like this is not a common practice, but some people think about it (especially those that don't have access to a store with these figures), and this data is meant to show that buying random packs to get a complete set is counterproductive.

Edited October 21, 20168 yr by BrickHat