Sneaking into Enemy Territory

Geometry Level 4

Lake of radius 1 mile in enemy soil where you are dropped

You are a Special Forces soldier trained to penetrate into enemy territories. On a pitch-dark night, a small plane secretly drops you at a point on a lake, which is circular and has radius 1 mile. But with all the heavy equipment needed for your mission, you can only swim up to 1 mile.

As soon as you are in the water, you start swimming in a direction chosen uniformly at random due to zero visibility. But depending on where you were dropped in the lake relative to the shore, the probability $p$ of your survival (i.e. getting to the shore) varies. More specifically, its lower and upper boundaries are as follows: $a<p<b.$

What is $b-a?$

Note: You were dropped at a point other than the center of the lake.

\frac16

\frac56

\frac23

\frac13

1 solution

Jimin Khim Staff
Apr 21, 2018

$The closer you are dropped to the center of the lake, the closer your probability of survival approaches $\frac12$ from the upper boundary of $\frac23.$$ The closer you are dropped to the center of the lake, the closer your probability of survival approaches $\frac12$ from the upper boundary of $\frac23.$

As shown in the diagram, suppose you were dropped very close to the shore. Then, unless you chose the range of directions indicated by the red arrows, you can safely reach the shore. So, your chances of survival in this case are very close to $b=\frac23.$ Now, note that as your landing point on the water approaches the center of the lake, the range of directions leading to "drowning" increases from $\frac13$ very close to $\frac12.$ So, the lower boundary is $a=\frac12.$ Therefore, $b-a=\frac23-\frac12=\frac16.\ _\square$

Sneaking into Enemy Territory

1 solution

0 pending reports