Just in case you don’t have a seven-year old boy in your house (in which case this will be obvious) a well-known brand of breakfast cereal here in NZ is currently coming with All-Blacks stats cards. Perfect for finding out your favourite rugby player’s height, number of caps, and how much they can eat for breakfast. Buy a family pack and you get four cards. We now have way more whole wheat biscuits in our house than we are likely to get through before Christmas, much to the delight of the cereal company no doubt. What great advertising.
Anyway, there are 40 different cards (current All Blacks plus some “all time greats”) and an obvious question that sprung to the mind of my seven-year old was “Wouldn’t it be amazing if I opened the pack and got all three Barretts (Jordie, Scott and Beauden) in the same pack. What are the chances of that?”
Well, the calculation isn’t that difficult if you make some assumptions. Let’s assume that each card could equally likely be one of the 40 possibilities. So no-one checks whether you are getting two cards the same, and no card is deliberately produced in smaller numbers to encourage yet more buying of breakfast cereal in an attempt to find the final elusive All Black. (Being cynical, I have my doubts on the validity of this last assumption.) That means the chances of getting a Barrett with the first card are 3 in 40. The chances of getting a different Barrett with your second card are 2 in 40 (because who wants two of the same?) and the chances of getting the third remaining Barrett with your third card are 1 in 40. If you succeed in that, we don’t care who we get with our fourth card. So the overall probability of pulling three different Barretts out with our first three cards are 3/40 times 2/40 times 1/40 which is 6/64000 or 0.00009375. About one chance in ten thousand.
BUT we have more options than that. Rather than pulling out Barrett-Barrett-Barrett-AllBlack in that order, we could pull out Barrett-Barrett-AllBlack-Barrett or Barrett-AllBlack-Barrett-Barrett or AllBlack-Barrett-Barrett-Barrett (where the three Barretts include Beauden, Jordie and Scott, but in any order. Are you following this?) So there are four times the possibilities than calculated above, in other words the chances of getting all Scott, Beauden and Jordie all in a single four-card pack is about four in ten thousand. Given the number of family packs of cereal bought in NZ a week, I suspect there are some lucky children who have had the Barrett trio all turn up to breakfast at once.
How did Benjamin do? Just a single Barrett: Beauden.
How many different combinations of All-Blacks could you get in a four-card pack, then? Here the numbers begin to get very large indeed. If we don’t worry about repeats, then there are 40 possibilities for each card, so on the face of it one might imagine the answer is one in 40 x 40 x 40 x 40 2.56 million. But, it’s more complicated than that. Which order do those cards come in? If we count Bridge, B. Barrett, B. Smith and Tuipulotu in that order as being equivalent to B. Barrett, Tuipulotu, Bridge and B. Smith in that order, then there are many combinations that are equivalent, so it’s not as big as 2.56 million. But it’s still pretty large. And that’s only with 4 cards and 40 All-Blacks. What if we went through the all-time All-Black list (past 1000 now) and gave out 100 cards with each cereal box. I’m not going to do the calculation, but the number of possibilities will be staggering.
This is essentially what the idea of entropy, much used in thermodynamics in physics and chemistry, comes down to. The more possibilities for something happening, the more likely it is to happen. Entropy is a measure of disorder in a system, and high entropy states are more likely than low entropy states. Why are chemical reactions normally exothermic (giving off energy to the surroundings – e.g. burning petrol)? It’s because the number of ways of sharing the extra heat energy around zillions of gas molecules in the surroundings is truly enormous. Just like like the number of ways of arranging two hundred miscellaneous objects throughout 7-year-old’s room in no particular order far exceeds the number of ways of arranging them in a tidy, sensible order, physical systems tend to move towards states of disorder. That’s entropy for you.