Numb3rs 402: Hollywood Homicide

Don and the gang are at it again; this time investiagting the murder of a young girl in a famous Hollywood star's bathtub. As usual, Charlie does his part to help unlock the keys to the mystery. First, he uses Snell's Law (among other things) to identify the victim from a photo. Second, he uses Archemdies' Principle to determine the size of the killer. Finally, he uses some game theory to figure out the motive for the killing.

Snell's Law

Snell's Law has to do with how light refracts (or bends) when it meets makes the change from passing through air to passing through water (or any other medium).

Activity: In this activity, we're going to see just how much water can bend light. To start, fill a glass with water and drop a penny in the bottom. Now, only looking straight down from above the glass, try to touch the penny with the point of a pencil. Not too hard, right? Now, tilt your head so that you are only looking at the penny from the side of the glass. Again, try to touch the penny with the tip of the pencil. Not as easy as before, eh? Just to prove that this activity isn't completely contrived, imagine now that you are a bear, the glass of water is large lake or river, the penny is a fish, and the pencil is your paw. Understanding how to hit the penny (or fish) is a bit more important when it leads to one of your few sources of protein.

Let's state Snell's law and see how it applies to our penny/fish situation:

Snell's Law: Let n1 and n2 be the refractive indicies of air and water respecitively. Let &theta1 be the angle of incidence and &theta2 be the angle of refraction. Then n1sin&theta1 = n2sin&theta2.
The following picture should help make Snell's law clear:

Let's now apply what we have learned to our penny problem:

In the first attempt, we were looking straight down at the penny. Thus, our angle of incidence was zero. That is, &theta1=0. Now, since we assume that n1 and n2 are non-zero, we get that 0= n1sin 0= n2sin&theta2, and so &theta2=0 as well. In other words, the light didn't bend at all, and we were able to touch the penny easily.

When we looked through the side of the glass however, we got a positive value for &theta1 and thus, the penny looked like it was somewhere it wasn't.

Game Theory

After using Snell's law and facial recognition software to identify the victim, Charlie realizes that he can use some game theory to help determine the motive for the crime. In particular, he uses the concept of risk and response.

A quatifiable way to illustrate this is by thinking about the game of basketball. Consider the great center Shaquille O'Neill. He has made an estonishing 58% of the shots he has taken in his career. However, he has made an abismal 52% of the freethrows that he has attempted. By comparison, Michael Jordan made only 50% of the shots he took in his career but 84% of his freethrows.

Now suppose you have the unenviable task of guarding either of these to legends. Should you rely on your defence, or foul so that he has to shoot freethrows? The answer, of course, lies in the math.

If you decide to defend Shaq, he is going to make 58% of his shots. If you decide instead to foul him when he shoots, the chances that he hits both freethrows is (.52)2, or 27%. So you clearly run a much greater risk trying to play defence. Thus, your expected response would be to foul him. Of course, your strategy chages when when we add in the rule that you are only allowed to foul him six times before being removed from the game, but we won't worry about that for now.

What to do against Michael Jordan, however, is a different story. He has a (.84)2 or 71% chance of making both freethrows if you decide to foul him, as opposed to only a 50% chance of making the shot in the first place. Thus, you risk much more by fouling him than by defending against him. Let's just hope that your team gets the rebound the times that he misses!