

Puzzle: The Road Less Traveled - retupmoc01
https://www.quantamagazine.org/20150903-the-road-less-traveled/

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dan_s_00
I'm trying to get better at prob&stats, so this was a good puzzle for me. I'm
not certain I have the right answer, but it seems straightforward enough after
thinking it through.

Spoilers follow!!

For question one: A and B will each have, on average, 100 drivers. Suppose we
also know the std. deviation from that average. Then the more popular road can
be expected to have (100 + stddev) drivers, and the less popular road (100 -
stddev) drivers. (This doesn't change the expected # of drivers for each road,
since either road has an equal chance of being the more popular one).

The binomial distribution
([https://en.wikipedia.org/wiki/Binomial_distribution](https://en.wikipedia.org/wiki/Binomial_distribution))
("given n coin flips each with chance p of being heads, what's the probability
that k of them were heads?") perfectly models the road choice, and its
variance (stddev^2) is conveniently available in closed form: np(1-p), which
given n=200 cars and p=0.5 is 50. So the expected number of drivers on more
popular road is 100+sqrt(50), and the less popular road 100-sqrt(50).

For question two: the article gives away the trick, for the most part -- we
count the number of cars each driver observes, not the number of cars each
road observes. There are 200 drivers, and (100+sqrt(50)) of them observe the
other (100+sqrt(50)) drivers on the more popular road, and similarly on the
less popular road. So the expected number of cars a _driver_ will see is 1/200
* (100+sqrt(50))^2 + (100-sqrt(50))^2, which works out to 114.6.

TLDR: If there are two equivalent roads and you and 199 other drivers make an
independent coin flip on which road to take, you should expect to observe 115
cars on the road you choose.

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barkingcat
Selection bias is a thing but I don't understand the core analogy of picking a
road. Every single time I pick a road less traveled I end up driving alone
with not a car in sight for hours. Until i get scared and retrace my path to
where there _are_ people.

maybe if you want a road that is not congested you pick both the road _and_
the destination less travelled. nobody wants to go there, ergo you get the
road alone.

