Probability in the Pool: August 19, 2013

When my mom had a catastrophic stroke, I returned to my home town and found a pool in which to swim.  I swam mostly alone in a pool that was 50 meters by 25 meters.  Sometimes the lane lines were set to allow swimmers to swim "short course" (25 meters) or "long course" (50 meters).

On one of the short course days, I was alone in the pool.  There were 16 empty lanes.  I chose lane 8, one of the middle lanes.  I swam down-and-back (50 meters) in about 45 seconds.  This became a deliciously long swim as I moved back and forth alone in this pool with my own personal lifeguard.  Then, suddenly, I was jerked out of my fraction-calculating delerium (refer to previous post about Fractions in the Fast Lane) when waves overtook me.  A man dropped himself and his large belly covered in fur into the lane next to mine at precisely the moment I was at his end of the pool.

Ah, I thought.  A probability problem.   What is the probability that he would (a) choose the lane next to mine and at the same time (b) choose to enter the water during the roughly 7 seconds that I am vulnerable to the tsunami he created at the near end of the pool.  And is this probability small enough that I should think this individual inconsiderate?

On one of the long course days, I was again alone in the pool, now well spoiled and feeling like the Queen of Sheba in her own blue-glass lake (with lane-lines and a black line at the bottom.)  There were 9 lanes. I chose the middle lane.  I would swim 50 meters to the far end and 50 meters back, giving me new distances and fractions to consider.  I swam the same rate as in the short course setting in the pool.  Again, as I swam a deliciously long swim with new numbers flowing along side of me, another man, a more narrow man, entered the pool.  Now then, many of us in this world like to be individuals; we value our ability to make choices and to be unique.  This man decided to swim a uniquely different way in this pool.  He wanted to swim 25 meters, not 50 meters as the pool was set.  So he did, weaving his way across the pool, below the lane lines and intersecting my path perpendicularly.  My interest became piqued.  As I swam, I watched the red clocks surrounding the pool and observed that he swam breast-stroke quite regularly, finishing 50 meters (down and back, under the lane-lines) in about 100 seconds.

Ah, I thought. Another math problem. If we both start at the same place (let's say a corner of the pool, just to make this conceptually more straight-forward) and swim in paths perpendicular to each other, when (if at all) will we collide?  If we both leave one end of the pool at the same time and swim at our own constant paces, when (if at all) will we collide?  If a moment during our infinite-length swim is chosen randomly, what is the probability that we will be in a collision at that moment?  Consider that it would be about 7 seconds for me to be in his way; he'd be in my way for 1/9th of the way across the 25 meter pool.