Skillful Skiing

You and your skiing opponent will complete two runs down the mountain, and the times of your runs will be added together. You and your opponent have the same, identical, independent normal probability distribution of finishing times for each run.

Given that you are ahead after the first run, what’s the probability that you will still be ahead after the second run and earn your gold medal?

Let your two random run completion times be denoted by $X_1$ and $X_2,$ and your opponent's $Y_1$ and $Y_2.$ Let $Z_1$ and $Z_2$ denote the difference between your run time and your opponent's. Then regardless of the initial mean and standard deviation of $X_1,$ $X_2,$ $Y_1,$ and $Y_2,$ $Z_1$ and $Z_2$ will be identical, independent normal variables with mean zero and some irrelevant but identical variance. Without loss of generality, we may in fact assume that the $Z_1$ and $Z_2$ are unit normals (with variance $1$).

Then the answer to the original question is given by $$\mathbb{P} \{ Z_1 + Z_2 \geq 0 \mid Z_1 \geq 0 \} = \frac{\mathbb{P} \{ Z_1 + Z_2 \geq 0, Z_1 \geq 0 \}}{ \mathbb{P} \{ Z_1 \geq 0 \}}.$$ Geometrically, the set $\{ Z_1 + Z_2 \geq 0, Z_1 \geq 0 \}$ can be written in polar coordinates as $\{ (\rho, \vartheta) : -\frac{\pi}{4} \leq \vartheta \leq \frac{\pi}{2} \},$ whereas the set $\{ Z_1 \geq 0 \}$ is given by $\{ (\rho, \vartheta) : -\frac{\pi}{2} \leq \vartheta \leq \frac{\pi}{2} \}.$ So by transforming to polar coordinates, we get the straightforward $$\mathbb{P} \{ Z_1 + Z_2 \geq 0, Z_1 \geq 0 \} = \int_0^\infty \rho e^{-\rho^2 / 2} \,d\rho \int_{-\pi/4}^{\pi/2}\, \frac{d\vartheta}{2\pi} = \frac{ \frac{3\pi}{4} }{ 2\pi} = \frac{3}{8}$$ and $$\mathbb{P} \{ Z_1 + Z_2 \geq 0, Z_1 \geq 0 \} = \int_0^\infty \rho e^{-\rho^2/2} \, d\rho \int_{-\pi/2}^{\pi/2} \, \frac{d\vartheta}{2\pi} = \frac{\pi}{ 2\pi} = \frac{1}{2}.$$

Put altogether, you get a conditional win probability after the first run as $$\mathbb{P} \{ Z_1 + Z_2 \geq 0 \mid Z_1 \geq 0 \} = \frac{3 / 8}{1 / 2} = \frac{3}{4}.$$ Of course, all of this argumentation leaves open not just superliminal but also time-traveling ski runs, so all in all it would be must-see but also potentially not-able-to-be-seen TV!