A friend who struggles with road rage recently asked me how I keep my cool with all the idiots on the road. I explained that I try to think about it mathematically:
$1)$ I assume I'm not a perfect driver, but I'm a "good" driver. Let's be VERY generous and assume that $95\%$ of the time, I am acting completely safely and considerately, and only $5\%$ of the time am I ever distracted, hurried, selfish, tired, or impaired in any capacity.
$2)$ Let's be even more generous and assume EVERY other driver is also "good".
Then I told him to imagine the number of cars he encounters every day in his commute in our small city. In that many cars, even if they were all good drivers, your chances of encountering SOMEONE on their $5\%$ moment would be pretty good. So it's illogical to ASSUME they are idiots, or whatever judgmental pejorative you feel.
In a world with only good drivers, you would still probably encounter bad driving every day. And if you can forgive yourself for your $5\%$ moments, there is no reason not to forgive the other drivers for theirs. This helped him not take it so personally.
My question is how would you express this problem mathematically?
Given only good drivers, how many cars would I have to encounter before I'm certain to encounter an otherwise good driver having a lapse in judgment?
Well, you would never be certain that you will definitely encounter someone having a bad day. However, let $P_n$ be the probability that you encounter someone driving poorly when you see $n$ drivers. Then
$$P_n=1-(.95)^n$$
If we want to know how many cars until we have a $50\%$ chance of seeing someone driving poorly, then we solve
$$.5=1-.95^n\Rightarrow n=13.5134$$
Since $n$ has to be an integer, you would have a greater than not chance of seeing someone driving poorly if your commute passed $14$ other cars. It's a wonder that we're all not dead every time we get in a vehicle.