We Ran The Data

Perfect timing!

We were out eating dinner tonight and we were discussing ACC basketball.  I was out numbered; my wife and TR are Duke fans while I’m a Carolina fan – daughter is a Tarheel but she was chillin’ with some friends.

Anyway, the conversation eventually landed on free throws and the difference between the One and One vs. the Two Shot Foul.

My question to TR was this, “Which situation results in more points, the One and One or the Two?”

He answered, “It depends on how well you shoot.”

Enter, “We Ran the Data”!

Of course it doesn’t matter how well you shoot, it matters how each condition is built.  When fouled there are two manners in which basketball players are awarded free throws – the One and One and then the Two Shot Foul.  In the first case, a player takes a free throw, and if he makes the first, he is awarded another shot.  In the two shot foul, the players gets two shots either way.

So, which is better and by how much?

Simple probability.

The probability of two events occurring is the product of each thing occurring in the first place. Think coin toss: 50% chance tails, 50% chance heads.  What is the probability of flipping heads twice?  Well, it’s 50% x 50% or .5*.5 = .25 or 25%.

So, free throws and Duke.  Take Grayson Allen, please.  Take Grayson Allen.  Hahaha.

Okay, the young man shots free throws at an 88% clip, so, on any given scenario, what are the odds that he scores 0, 1 or 2 points per foul?

First the One and One.

On the first shot, there are two outcomes – a made shot and then a miss.  The chance that Mr. Allen hits his first free throw is 88%, the chance that he misses is 12%.  Therefore, we know what the chance that he scores 0 points on a one and one is; 12%.  Now onto the remaining two possible outcomes.  So, we know if he misses his first, a 12% chance, he’s done, but what if he hits the first?  Well, he gets another shot.  He now has another 88% chance of hitting the second.  So, knowing he hit the first shot, he’s in a situation where he has an 88% chance of scoring TWO points from the foul and a 12% of scoring only one.  Using the rule above – the probability of two events occurring we can see that his chance of hitting two is 88% times 88% or .88*.88 = .77 or 77%.  The chance that he hits the first and whiffs the second is the percent chance of making a free throw, 88%, multiplied by the chance of missing a free throw, 12%.  This is 11%.

We have it.

The chance of scoring 0 points is 12%.  This is accomplished by missing the first shot.
The chance of scoring 1 point is 11%.  This is accomplished by making the first shot and then missing the second.
Finally, the chance of scoring 2 points is 77%.  And this is accomplished by making both shots.

A quick check confirms our math; 12% + 11% + 77% = 100%.

Now the Two Shot Foul.

A slight change in the rules.  Independent of the first shot, a player is guaranteed two shots.

Now the numbers change, and we see that scoring 0 points requires a player to miss two shots.  In Grayson’s case, that looks like 12% times 12% or 1%.

To score only 1 point, a player must hit and miss OR miss and hit.  For each, Grayson has a 88% x 12% and a 12% x 88% or an 11% for each case; hit and miss or miss and hit.

To score 2 points he has to make two shots; 88% times 88% = 77%

Adding them up:

0 points = 1%
1 point = 11% + 11% = 22 %
2 points = 77%

Check your math by adding those eventualities up and you get 100%.

Now, the last step.  Knowing that your shooter is an 88% free throw shooter, how many points does he score, on average, if he takes 100 One and Ones vs Two Shots?

Easy:

One and One:

12 trips yield 0 points
11 trips yield 1 point
77 trips yield 2 points

Total points in 100 trips is 165 divided by 100 is 1.65 points per trip.

Two Shot:

1 trip yields 0 points
22 trips yields 1 point
77 trips yield 2 points

Total points in 100 trips is 176 divided by 100 is 1.76 points per trip.