Written by Colin+ in probability, sport.

There are six minutes to play in the last Autumn international, and Australia are leading Wales by 30 points to 26. Australia, however, have just conceded a penalty in front of the posts, leaving the Welsh captain, Sam Warburton, with a dilemma: should he kick at goal (and take a virtually-guaranteed three points), or kick for position, giving Wales the chance of five - or even seven - points from a try?

That's a hard problem, as it stands: you don't know how likely Wales are to score a try - or to score again if they take the easy three points. You also don't know how likely Australia are to score on the counter-attack, and kill the game off. So let's simplify things - and use some of the game statistics to estimate the probabilities.

Assumption #1 is that the kick is so easy, no self-respecting test rugby player would ever miss it, so kicking for goal would make the score 29-30 with five minutes to play.

Wales had scored two tries in the first 74 minutes, meaning (if they were to carry on scoring at the same rate1 ) they would expect to score about 0.13 of a try in the remaining five. They had also scored six penalties - and would expect to score 0.4 of a penalty in the last five minutes.

Australia, meanwhile, had scored three tries (expect about 0.2 of a try before the ref blows up) and three penalties (about 0.2 of a penalty in the time that's left).

Assumption #2 is that the probabilities of any of those things happening are roughly the same as those numbers - i.e., there's a 13% chance Wales will score a try, and a 20% chance Australia will score a penalty; I'll also assume there's only time for one of these to happen.

That means, 53% of the time, Wales would score again and win the match; in the remaining 47% of cases, Australia would either hang on or extend their lead.

Assumption #4 is that one of two things will happen if Wales kick for territory: either they'll score a try, or play will break down. In either case, it'll take about the same time as kicking for goal - and then we'll have five minutes of regular play, just to reuse the analysis from earlier.

Trouble is, we don't have any information on how likely Wales are to score a try by kicking for the corner. We'll just have to call it $p$.

Assumption #5 is that any try would have a two-in-three chance of being converted - although this doesn't make any real difference to the analysis.

So, we have three scenarios:

1) Wales score and convert a try (probability $\frac {2p}{3}$). This will put them three points ahead. Looking at the scores from earlier, Australia would have a 20% chance of a tie (from a penalty) and 20% of winning (by scoring a try). Wales have a 60% of holding on in this scenario, and 20% of a tie.

2) Wales score a try but miss the conversion (probability $\frac{p}{3}$). This puts them one point ahead, in which case Australia would win with any further score - a 40% chance. Wales have a 60% chance of holding on.

3) Nothing comes of the chance and Wales remain four points behind (probability $(1-p)$. They would then need a try to win - a probability of 13% ((In likelihood, this would be higher, as they would decline the chance to kick at goal... but let's not make it any more complicated).

So, Wales's chances of winning if they kick for position would be $0.6p + 0.13(1-p)$, ignoring the chance of a tie. Nobody wants a tie.

So, Warburton should take the risky option if that win probability is higher than 53%, depending on how he assesses $p$:

$0.6p + 0.13(1-p) > 0.53$. which works out to:

$0.47p > 0.4$, or $p > \frac{40}{47} \simeq 0.85$

Wales should only kick for the corner - under this analysis - if they think they'll score a try from that position five times out of six. To me, that seems absurdly high.

In the event, Wales kicked for position, but nothing came of the resulting move. Australia held on to maintain their unbeaten streak against Wales... for now. Warburton was roundly criticised for his risky decision - although I imagine he'd have been hailed as a hero if it had come off.

Depending on how he assessed the situation - most critically, Wales's chances of scoring a try in the last few minutes against 14 men - Warburton's decision could have been right or wrong. Under the model I've shown here, it was wrong; but there are many other possible models. This one is certainly wrong, but possibly useful.

- This isn't a brilliant assumption: Australia's Wade Cooper had just been sin-binned for a professional foul, and scoring rates tend to increase towards the end of a game. However, for the sake of keeping the analysis relatively simple, I'll run with it. [↩]