### What's the Value of a Win?

In a previous entry I demonstrated one simple way to estimate an exponent for the Pythagorean win expectation. Another nice consequence of a Pythagorean win expectation formula is that it also makes it simple to estimate the run value of a win in baseball, the point value of a win in basketball, the goal value of a win in hockey etc.

Let our Pythagorean win expectation formula be $w=\frac{P^e}{P^e+1},$ where $$w$$ is the win fraction expectation, $$P$$ is runs/allowed (or similar) and $$e$$ is the Pythagorean exponent. How do we get an estimate for the run value of a win? The expected number of games won in a season with $$g$$ games is $W = g\cdot w = g\cdot \frac{P^e}{P^e+1},$ so for one estimate we only need to compute the value of the partial derivative $$\frac{\partial W}{\partial P}$$ at $$P=1$$. Note that $W = g\left( 1-\frac{1}{P^e+1}\right),$ and so $\frac{\partial W}{\partial P} = g\frac{eP^{e-1}}{(P^e+1)^2}$ and it follows $\frac{\partial W}{\partial P}(P=1) = \frac{ge}{4}.$ Our estimate for the run value of a win now follows by setting $\frac{\Delta W}{\Delta P} = \frac{ge}{4}$ giving $\Delta W = 1 = \frac{ge}{4} \Delta P.$ What is $$\Delta P$$? Well $$P = R/A$$, where $$R$$ is runs scored over the season and $$A$$ is runs allowed over the season. We're assuming this is a league average team and asking how many more runs they'd need to score to win an additional game, so $$A$$ is actually fixed at $$L$$, the league average number of runs scored (or allowed). This gives us $1 = \frac{ge}{4} \Delta P = \frac{ge\Delta R}{4L}.$ Now $$L/g = l$$, the league average runs per game, so we arrive at the estimate $\Delta R = \frac{4l}{e}.$ In the specific case of MLB, we have $$e = 1.8$$ and $$l = 4.3$$, giving that a win is approximately $$\Delta R = 9.56$$ runs.

Bill James originally used the exponent $$e=2$$; in this case the formula simplifies to $$\Delta R = 2l$$, i.e. we get the particularly simple result that a win is equal to approximately twice the average number of runs scored per game.

Applying this estimate to the NBA, a win is approximately $$\Delta R = \frac{4\cdot 101}{16.4} = 24.6$$ points. Similarly, we get the estimates for a win of 4.5 goals for the NHL and 5.1 goals for the Premier League.

1. I think you've assigned the incorrect goals/win to the wrong league. NHL I think is 5.1 and Premier League is 4.5. Thanks for sharing!!

2. Thanks, I believe you're right!

### A Bayes' Solution to Monty Hall

For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem. Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way.

Standard notation:

probability of A given B is written $$\Pr(A \mid B)$$
probability of B is written $$\Pr(B)$$

Bayes' Theorem:

Using the notation above, Bayes' Theorem can be written: $\Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)}$Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.

Assume we pick Door 1 and then Monty sho…

### Solving a Math Puzzle using Physics

The following math problem, which appeared on a Scottish maths paper, has been making the internet rounds.

The first two parts require students to interpret the meaning of the components of the formula $$T(x) = 5 \sqrt{36+x^2} + 4(20-x)$$, and the final "challenge" component involves finding the minimum of $$T(x)$$ over $$0 \leq x \leq 20$$. Usually this would require a differentiation, but if you know Snell's law you can write down the solution almost immediately. People normally think of Snell's law in the context of light and optics, but it's really a statement about least time across media permitting different velocities.

One way to phrase Snell's law is that least travel time is achieved when $\frac{\sin{\theta_1}}{\sin{\theta_2}} = \frac{v_1}{v_2},$ where $$\theta_1, \theta_2$$ are the angles to the normal and $$v_1, v_2$$ are the travel velocities in the two media.

In our puzzle the crocodile has an implied travel velocity of 1/5 in the water …