collect 'em all - what are the odds?

June 2, 2006 9:24 AM

4 different models of a toy are sold in identical, sealed boxes. Assuming the 4 different models are produced/sold/distributed in equal numbers, etc..., what are the chances of buying only 4 boxes, chosen at random (at different times, even), and getting all 4 different toys?

It happened to me. When I got the last one I felt like I had won the lottery. Seems like the chances would be pretty slim - I know this is probably about as basic as probability questions go, but I'm no mathematician. And if you're interested, the toys in question were Gary Baseman's collectible vinyl Fire Water Bunny series:

posted by ab3 to grab bag (12 comments total)

Is there any way to tell if an employee who sold them to you/packed them for shipment could tell what was in each box? You could have been the recipient of some kindness to a collector...
posted by agregoli at 9:25 AM on June 2, 2006

4/4 * 3/4 * 2/4 * 1/4 = 9.375%.
posted by kcm at 9:27 AM on June 2, 2006

Looking at it another way, you could rephrase the problem as: given a four digit number where all the digits are 1, 2, 3, or 4, what percentage of numbers have one of each digit?

There are 4*4*4*4=256 combinations in total, and 4*3*2*1=24 combinations where the digits are unique, so the odds are 24/256 = 0.09375, or approximately 1 in 10.

On preview: what kcm said.
posted by Armitage Shanks at 9:34 AM on June 2, 2006

It's actually slightly higher than kcm or Armitage Shanks suggest, though so slightly that it's probably negligible. It's slightly higher because if you bought model A, then there's one less model A for sale so you're slightly less likely to buy it next time, increasing the chances you get a new one.
posted by louigi at 9:46 AM on June 2, 2006

see? i knew it was easy. and btw, agregoli, the toys come in opaque mylar bags inside the identical boxes - and i chose each box myself, at random from a stack of maybe 30 or 40 on a shelf. so human assistance wasn't a factor.
posted by ab3 at 9:46 AM on June 2, 2006

Alright, just checking. That would definitely skew the statistics a bit if someone did it for you.
posted by agregoli at 9:49 AM on June 2, 2006

I'd just like to chime in that I approve of the toy choice. I have the "naked" version sitting on my desk right now.

:-)

On topic, 10% isn't that terrible a chance, overall. Intuitively I'd have thought it was less than that.
posted by griffey at 10:03 AM on June 2, 2006

Just curious: why would you buy the toys without knowing the contents were, with the possibility that you'd get a duplicate?
posted by Big Fat Tycoon at 11:53 AM on June 2, 2006

They seem to do this all the time with collectables, and I don't understand it either - you're just out of luck if you get two. Seems endlessly frustrating.
posted by agregoli at 11:59 AM on June 2, 2006

I guess the follow-on question would be:

How many does the "average" person who wants all four, end up actually buying?
posted by vacapinta at 12:36 PM on June 2, 2006

Big Fat Tycoon -

um.. because you want the toys?
posted by ab3 at 1:41 PM on June 2, 2006

There are several sellers online that will open the box so you can choose which ones you want...I've seen the option to buy them opened or unopened. I don't know that they have those particular figurines you pictured though. Or, check eBay.
posted by IndigoRain at 7:41 AM on June 3, 2006