By Brendan Kent

Keeper penalties are quite simply the best. First of all, if a keeper is taking a penalty we are most likely in sudden death penalties of a knockout match. Pretty exciting to begin with. Now add the fact that the players taking the penalties are those whose job is the opposite of goal scoring.

If you watched the Portland Timbers vs. Sporting Kansas City knockout round playoff game on October 29^{th}, you know what I’m talking about. After 10 rounds of penalties that saw Sporting’s potential winner bounce off both posts to stay out, Portland keeper, Adam Kwarasey, stepped up the spot and slotted past Jon Kempin. Kwarasey then saved Kempin’s ensuing shot to send the Timbers through.

Full disclosure: I work for the Timbers as the Club’s Soccer Data Analyst so I’m still riding the high from that game. But, quite frankly, I probably would’ve enjoyed watching this particular shootout more as a neutral (fewer heart stoppages, etc.). There’s something very special about keepers taking penalties and it’s something I always hope for when watching a knockout match as a neutral. But what are the chances of actually seeing it?

First we make an important assumption. It’s not always the case that the keeper is the last penalty taker in the lineup and the keeper could, in fact, be very high up in the lineup. But for the sake of simplicity, we’re going to focus on the likelihood of a shootout going to 11 rounds. This can also be thought of as the probability that both keepers will definitely take a penalty at some point – whether or not the penalty is taken in the 11^{th} slot.

To find the probability of a shootout going to at least the 11^{th} round, I used Monte Carlo simulation, simulating 100,000 shootouts and finding the percentage that went to at least the 11^{th} round. Again, we need to make several assumptions. The first is the rate at which penalties are converted. During the 2014 World Cup, FiveThirtyEight published a piece in which they noted that the success rate of a penalty (shootout penalties, not penalties as a result of a foul) in major tournaments is circa 75%. This obviously changes depending on quality of the players, but it’s a good estimation. We also will assume (for now) that both teams convert penalties at the same rate. Finally, we will assume that each shot is independent, that is, each shot is an independent Bernoulli trial with p = .75. Again, this is not a trivial assumption, as there are probably psychological aspects to a shootout that create dependencies between shots, but we’ll go with it.

Finally, the results. Simulating 100,000 shootouts with each team’s conversion rate at 75%, we get the following distribution…

The percentage of shootouts that go to at least 11 rounds is, therefore, 2.8%. A bit lower than us keeper-penalty fans might like – but then again rarity is what makes these events so exciting.

Now, the conversion rate itself is going to have an effect on the likelihood of an 11-round or more shootout. In theory, extreme conversion rates will mean longer shootouts – if the conversion rate is 1, an 11^{th} round is guaranteed, the same goes for a conversion rate of 0. Simulating 100,000 shootouts with different conversion rates between 0 and 1, we get the following graph that matches our intuition…

As the graph confirms, shootouts with 11 or more rounds are most likely to occur when the conversion rate is very high or very low.

Let’s tweak a couple things in the simulation. We now drop the assumption that both teams convert penalties at the same rate. Intuitively, this should lower the rate of shootouts that go to the 11^{th} round or later, as the better team should generally be able to finish off the weaker team in fewer rounds. Running the simulation again with Team 1’s conversion rate at 85% and Team 2’s conversion rate at 65%, the percentage of shootouts that go to 11+ rounds indeed falls 1% from our original 75%-75% simulation to 1.8%.

Essentially, we are most likely to see keepers take penalties (shootouts with 11 or more rounds) when both teams have similar conversion rates that are either very high or very low. So the probability of England keeper, Joe Hart, taking a penalty? Almost zero.

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