Marilyn vos Savant still has her column in Parade magazine, which introduced everyone to the Monty Hall Problem. Here's her latest (well, a repeat from 1995):
A schoolboy has a test coming up and so wants to study for it, but is conflicted, as there's a movie he wants to see playing at the local theater. He starts out toward the theater, but halfway there he has second thoughts and turns back toward home. But again, halfway home he has third thoughts and turns back again, this time toward the theater, but again, halfway there he turns back toward home, and this indecision continues in the same way with the same switching ad infinitum.
What does this lead to as to his location in the long run?
Let home be at x=0, the theater is at x=1.
His first few locations are:
0, 1/2, 1/4, 5/8, 5/16, 21/32, 21/64,...
Call these locations 0,1,2,...
f(n+1) = f(n)/2 if n is odd
f(n+1) = f(n) + (1-f(n))/2 if n is even
Simplifying:
f(n+1) = f(n)/2 + 1/2 if n is even
In the steady state, our hero will simply go back and forth between 2 locations.
Could the 2 locations be the same location? If so we would have:
x/2 = x/2 + 1/2 with no solution, so no.
Then find x where 2 steps return to the same spot:
x --> x/2 --> x/4 + 1/2
x = x/4 + 1/2 --> x = 2/3
So the schoolboy will end up going back and forth between locations 2/3 and 1/3
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Posted by Larry
on 2023-08-17 09:03:41 |
Solution
