My wife and I like to stay active, so one of our hobbies is amateur
cross-country running. We’re not very good, but we do like to run
in weekend charity races. We decided several years ago to favor
small-town races over downtown ones, and smaller races over larger
ones. We’ve been pleased with these rules: folks who put on
a race with only a hundred racers really appreciate your entry, and
will introduce themselves, and thank you for coming–and they really
mean it! The small races are often better organized, too.
The local Lions, Kiwanis, Red Cross, or Band Boosters want everything
to be perfect, so nobody goofs off. But best of all, sometimes
small races happen on the same day as big races. That means all
the superfast runners are off somewhere glitzy, and we trudgers have a
chance at 15 seconds of glory in the winners circle.
Saturday was just such a race. While about 1000 big deal runners were racing at the SBC Center with the San Antonio Silver Stars, 200 of us ran the Live Oak Lions 5K. It was hot and hilly, and timewise, neither my wife or I covered ourselves with glory. But as our friend Jose Iniquez
says, "You only need to run fast enough to win" and that’s what we
did: we both placed first in our respective age groups!
Afterwards, it occured to me that this situation was a wonderful setting for a probability puzzle:
weekends I like to run in amateur 5K cross-country races. I’m not
that good, so I seldom place in any of the big races, even in my age
group. In fact, of the 13 guys in my age group, I’m right in the
middle, usually ranked seventh. Lately I hit upon a strategy that
lets me place–in my age group–more often. Some weekends two different
races are scheduled simultaneously, and if I enter the smaller race
(the one with few runners), I have a better chance of placing. Assume
that the guys in my age group attend either of the two races with
probabilities proportional to the number of runners in each race.
Also assume that previous ranks determine who will win.
races will be held next Saturday, one with 200 runners and one with 400
runners, and I will attend the smaller one. What is the
probability that I will be one of the top 3 finishers in my age group?
two races are held next month, one with 250 runners and one with 750
runners, what is the probability that I will win my age group in the