# Pass the pigs – ad (almost) infinitum

I warn you, this post will make little sense to anyone who doesn’t know the game Pass the Pigs.

What do you do on a damp Easter Holiday Monday? (Not that the Holiday Monday bit makes much difference in our house…) Play Pass the Pigs of course. And what do you do when you’re a physicist who’s finished playing Pass the Pigs? You decide to work out the optimum tactics. It’s not just down to luck – the skill is to know when to ‘bank’ your points and when to keep continuing, risking a Pig Out that will lose your points for that go.  To optimize the strategy, you need to know the probabilities of getting the various combinations.

Fortunately there’s a seven year old in the house who is learning about percentages and plotting bar graphs and pie charts. So, one thousand, one hundred and eleven rolls later, here is the data we’ve accumulated  (percentages in brackets):

Pig out (pigs land on opposite sides – lose your points for the go): 217 times (20%)

Sider (1 point, pigs land on same side): 233 times (21%)

Trotter (5 points, one pig lands on its feet): 94 (8.5%)

Double trotter (20 points, both pigs land on their feet): 8 (0.7%)

Razorback (5 points, one pig lands on its back): 331 (30%)

Double Razorback (20 points, both pigs land on their backs): 104 (9.4%)

Snouter (10 points, one pig lands on its nose and two front legs): 34 (3.1%)

Double snouter (40 points, both pigs land in the snouter position): 5 (0.5%)

Leaning jowler (15 points, one pig lands balanced on its ear, snout and one front leg): 12 (1.1%)

Double leaning jowler (60 points, both pigs land a leaning jowler): 1 (0.09%)*

Razorback+trotter (10 points, one of each): 47 (4.2%)

Snouter+trotter: (15 points): 2 (1.8%)

Snouter+razorback: (15 points): 5 (0.5%)

Makin’ Bacon (pigs land touching each other – it loses all your points for the whole game): 18 (1.6%)

And there you have it. The next step is to work out the expected score for a roll. I’ll ignore the Makin’ Bacon bit (that makes it difficult). I’ll ask the question: am I better off rolling or not rolling?  Well, suppose I have a total of X points for my go. I have a probability of 0.20 of pigging out, which would give an expected contribution of -0.2 X.  I can add to that the expected contributions from all the other possibilities by multiplying their points by their probabilities. After a bit of calculator-work, this comes to 5.4.   So my expected gain for my next roll is -0.2X + 5.4.  For this to be positive (and thus on average increase my score) I need X smaller than 5.4 / 0.2, which is 27.  In other words, my optimum strategy, based on this analysis alone, might be to keep rolling till I have got to 27 or more points, then bank it.

There are other complicating factors though – such as your opponents score. The first to 100 wins, so you need to adjust your risk taking accordingly.

Is there any serious point to this? Well, it helps teach my child about probability, statistics and graphing data! There is a physics question too. Now, the pigs are not quite symmetrical, since their legs are in different positions on their left and right sides. This means there is no reason as such to assume that the chance of a single pig landing on its left side is the same as the chance of it landing on its right side. Thus Pig Out (pigs land on opposite sides) and Sider (pigs land on the same side) are not necessarily equally probable. The data shows they are pretty similar, but, perhaps, with 10,000 rolls (that will keep Benjamin occupied) there might be a difference start to show. It’s just a touch like CP violation in particle physics – where there might be subtle differences between a neutrino and an anti-neutrino (the two being not quite symmetric opposites of each other). It would take a monster-sized experiment to collect enough data to see this (neutrinos being hard to detect in the first place), but that’s just what the T2K experiment at Kamioka in  Japan is trying to do. And if it’s successful it could explain one of the big mysteries in physics – why is the universe dominated by matter, rather than having matter and antimatter in equal amounts.

Or, putting it another way, why do we have pigs but not anti-pigs?

*I cannot independently verify this. Benjamin insists he got one, but I didn’t see it. I have to say I’m a bit skeptical – the DLJ is the Royal Flush of Pass the Pigs. 60 points is far too few – It should be an instant win for the game (or perhaps for the next ten games).