i am reginald reagan aka RAGIN' RAYGUNS

Recently I went to a class and showed up late. I regretted my lateness, but I wondered, would leaving earlier really have been rational? I was taking a new route to class, and I didn’t know how long it would take. For all I knew, if I left earlier, I’d have been there way early and wasted my time.

So today I tried to think mathematically about this decision. If you leave early, you can guarantee that you arrive on time, but you might have to wait for the class to start. If you leave late, you won’t have to wait, but you take the risk of showing up late. So, when should you leave?

I modeled your arrival time as a normal distribution. The standard deviation of the distribution represents the randomness in the arrival time. The plot above shows that if you know your arrival time ±1 minute (red line), you can achieve a very high (99%) probability of showing up on time, and only get there 1 minute early on average. However, once you have an uncertainty of 5 minutes in your arrival time (cyan line), really guaranteeing that you show up on time requires getting there very early on average, like 8-10 minutes, which doesn’t seem worth it. However you can reduce the wait time to 5 minutes by accepting 15% chance of being late.

I had a very high uncertainty in my arrival time, probably like 5-10 minutes. Which I could use to argue that it was rational to take the risk of being late, but really I should have just used Google maps to get a prediction of how long it would take to walk there.

R code for the plot: 

plot(pnorm(0,mean=-seq(0,10,.01),sd=1), seq(0,10,.01), type='l',
     main="The price you pay for arriving to class on time",
     xlab="Probability of arriving on time",
     ylab="How early you are on average (in minutes)",
     col=rainbow(4)[1])
lines(pnorm(0,mean=-seq(0,10,.01),sd=2),seq(0,10,.01),
      col=rainbow(4)[2])
lines(pnorm(0,mean=-seq(0,10,.01),sd=5), seq(0,10,.01),
      col=rainbow(4)[3])
lines(pnorm(0,mean=-seq(0,10,.01),sd=10), seq(0,10,.01),
      col=rainbow(4)[4])
legend(.5, 10, c("sd=1 min", "sd=2 min", "sd=5 min", "sd=10 min"),
       rainbow(4), cex=1, bty='n')