How unlucky (or lucky) do you have to be to get a randomly assorted box of chocolates that contains only one type of chocolate?
People posted an unusual complaint to Cadbury’s Facebook page over the Easter break. The chocolate giant’s Favourites egg box comes with one large egg and a selection of eight chocolates from a possible five different varieties, but some customers were annoyed at getting too many of one type of chocolate.
One person complained about getting seven Cherry Ripes. Shelley Down from New Zealand wrote on the Facebook page: “Was super excited to watch my son open his Favourites Easter Egg but had a bit of a WTF moment when we realised there were seven Cherry Ripes and one Dairy Milk. I know Cherry Ripes get a bit of a tough time and you still need to force them upon us but the ratio seems a bit drastic!”
One customer was disappointed at not receiving any Caramellos or Flakes. Another posted a photo showing they had received only Crunchies.
Responding to posts, Cadbury maintained its chocolate addition process was random. So, assuming each chocolate type is added randomly, what is the probability of getting the combination people complained about? With the help of maths gurus Jack, from the University of Sydney, and Andrew, formerly at the Australian National University, we’ve crunched the numbers.
Seven Cherry Ripes – pretty unlucky, but could happen a few times if you bought everyone at a decent-sized football match a box of chocolates
With five possible chocolates in eight spots, there is a total of 390,625 outcomes. There’s 32 outcomes that match the criteria of seven Cherry Ripes and one other chocolate that’s not a Cherry Ripe, so the probability is 32 / 390,625, or about 1 / 12,207.
No Caramellos or Flakes – unlucky, but would happen occasionally
There’s 6561 outcomes that meet the criteria so the probability is 6,561 / 390,625, or about 1 / 60.
All Crunchies – this guy should do whatever the opposite of buying a lottery ticket is
There’s only one outcome out of the total 390,625 possibilities that could result in all Crunchies, so this guy is incredibly unlucky. Maybe he should trade a few for Cherry Ripes.
At least one of every chocolate – should happen occasionally
And here we have possibly the real issue for Favourites fans. With 126,000 outcomes containing at least one of each chocolate type, this should be the case around one-third of the time.
A more interesting aspect to the story is the possibility that Cadbury’s assignment of chocolates to boxes is not in fact random, and that there is bias involved. Maybe the size or shape of certain chocolates results in them clumping together, or something similar.
The way to test this would be to sample a large number of boxes to see if the real distribution deviates from the expected distribution. But I’m not sure if I could handle eating the thousands of chocolates required to get a statistically powerful sample size.
Thomas Lumley from the University of Auckland has also produced a simulation of 100,000 randomly assorted packets and suggests that an ordered approach to the solution is correct. This would put the probability for getting seven Cherry Ripes at between 0.6 and 1 out of 10,000, and the probability of getting at least one of every chocolate at around 30%.
Correction
Originally, this article had the odds for scenarios one, two, three and four as 4 / 495, 45 / 495, 1/ 495 and 35 / 495. This was incorrect, as it failed to consider that each unique combination of outcomes wasn’t equally likely. The odds have been updated above to show the correct probabilities. We consulted several maths authorities to work out these probabilities, but if we have something wrong please let us know at nick.evershed@theguardian.com
