This week's Riddler Express is, inevitably, about the World Cup again. It's one of those that begs to be solved empirically (which I've done here: https://github.com/johnfaben/riddler/tree/master/Penalties), but is probably also relatively easy to solve exactly, so I'll give that a go below.

This year’s World Cup has been chock full of exciting penalty shootouts. Historically, about 75 percent of soccer penalty kicks are successful. Given that number, what are the chances that a shootout goes past its fifth kick for each team and into theeven more excitingsudden-death portion of the penalty-kick period?

0Here's my first try. There are three ways for a shoot-out to end early. After three penalties, when one team is 3 ahead, after four penalties, when one team is 2 or more ahead, or after 5 penalties, when one team is 1 or more ahead. It ends after three about 0.7% of the time (one team makes all three of their 3/4 shots, and the other team misses all three, and 3*3*3/4*4*4*4*4*4 = 27/4096).

It ends after four if one team is 2 or more ahead after the first penalty, but was not 3 ahead after the fourth. This is... wait, this is going to get tricky. Why am I bothering to consider how many penalties are actually taken?!? That will lead to some horrible conditional probabilities, and all I really care about is whether it is over after 5 penalties.

Here's my second try. There are 6 ways for a shootout to go to sudden death. The teams score 0 each, 1 each, 2 each, 3 each, 4 each or 5 each.

The probabilities of these outcomes are:

0 each:

((1/4)^5)^2 = 1/1048576 ~= 0.00001%

1 each

(5C1 *3/4*(1/4)^4)^2 = 225/1048576 ~= 0.002%

2 each:

(5C2 * (3/4)^2 * (1/4)^3)^2 = 8100/1048576 ~= 0.77%

3 each:

(5C3 * (3/4)^3 * (1/4)^3)^2 = 72900/1048576 ~= 6.95%

4 each:

(5C4 * (3/4)^4 * 1/4)^2 = 164025/1048576 ~= 15.64%

5 each:

((3/4)^5)^2 = 59049/1048576 ~= 5.63%

So the final probability is 304300/1048576, or roughly 29%.

How does that match up with what I found with a simple simulation?

It ends after four if one team is 2 or more ahead after the first penalty, but was not 3 ahead after the fourth. This is... wait, this is going to get tricky. Why am I bothering to consider how many penalties are actually taken?!? That will lead to some horrible conditional probabilities, and all I really care about is whether it is over after 5 penalties.

Here's my second try. There are 6 ways for a shootout to go to sudden death. The teams score 0 each, 1 each, 2 each, 3 each, 4 each or 5 each.

The probabilities of these outcomes are:

0 each:

((1/4)^5)^2 = 1/1048576 ~= 0.00001%

1 each

(5C1 *3/4*(1/4)^4)^2 = 225/1048576 ~= 0.002%

2 each:

(5C2 * (3/4)^2 * (1/4)^3)^2 = 8100/1048576 ~= 0.77%

3 each:

(5C3 * (3/4)^3 * (1/4)^3)^2 = 72900/1048576 ~= 6.95%

4 each:

(5C4 * (3/4)^4 * 1/4)^2 = 164025/1048576 ~= 15.64%

5 each:

((3/4)^5)^2 = 59049/1048576 ~= 5.63%

So the final probability is 304300/1048576, or roughly 29%.

How does that match up with what I found with a simple simulation?

In 10000000 trials, draws happened 29.02% of the time

0 - 0: 0.00%

1 - 1: 0.02%

2 - 2: 0.77%

3 - 3: 6.95%

4 - 4: 15.64%

5 - 5: 5.63%

Not bad... not sure whether this should make me trust my maths more or trust my ability to code up tiny simulations more, probably a bit of both.

I would note that the Python version is much easier to adapt to different success rates... England had a 66% record in shoot-outs going into this World Cup, although they scored 4/5 against Colombia, bringing it to 70% overall, whereas Germany have scored 94%... a quick tweak to the script suggests that England would take Germany to sudden death just 21% of the time, losing in the first five kicks 73% of the, and winning just 6 times in 100. Luckily for England, Germany didn't make it past the group stages...

0 - 0: 0.00%

1 - 1: 0.02%

2 - 2: 0.77%

3 - 3: 6.95%

4 - 4: 15.64%

5 - 5: 5.63%

Not bad... not sure whether this should make me trust my maths more or trust my ability to code up tiny simulations more, probably a bit of both.

I would note that the Python version is much easier to adapt to different success rates... England had a 66% record in shoot-outs going into this World Cup, although they scored 4/5 against Colombia, bringing it to 70% overall, whereas Germany have scored 94%... a quick tweak to the script suggests that England would take Germany to sudden death just 21% of the time, losing in the first five kicks 73% of the, and winning just 6 times in 100. Luckily for England, Germany didn't make it past the group stages...