Consider these three zones starting from where
the elevator leaves floor 2 going up:
|2U 3U 4U 5U 6U 7D 6D 5D 4D 3D 2D 1U |
| Zone 1 |Zone2 |Zone 3 |
An elevator has 4/6 chance of being in zone 1 and
1/6 of being in 2 and 1/6 in 3.
There is a 50/50 race between an up and a down elevator
to get to floor 2 if and only if one is in zone 2 and the other in 3.
Here are the probabilities for the location of the two elevators and the chance up will arrive first:
Elevators in Zones ways/36 Pr(Up wins)
-----------------------------------------------------
3 & 3 1/36 1
1 & 3 8/36 1
2 & 3 2/36 1/2
1 & 2 25/26 0
So Up now gets prob ( 1 + 8 + 2 x 0.5)/ 36 = 10/36 = 0.2777...,
(way better than 1/6 = 0.1666... for the single elevator)
So with 1 elevator, on floor 2 prob(up) = 1/6, prob(down) = 5/6
Two elevators, prob(up) = 5/18, prob(down) = 13/18
here is the simulationord@rabbit-3 ~ % gamow
sim: prob. up = 0.278 analytic = 0.278
lord@rabbit-3 ~ %
lord@rabbit-3 ~ % more gamow.f
program gamow
implicit none
integer iseed,i,cnt
real rat,a,b,da,db,x
iseed=time8()
cnt=0
do i=1,1000000
a=rand()*12
b=rand()*12
da=10.-a ! unneeded!
if(a.gt.10)da=10.-a
db=10.-b ! uneeded !
if(b.gt.10)db=10.-b
if(abs(da).lt.abs(db).and.da.lt.0)cnt=cnt+1
if(abs(db).lt.abs(da).and.db.lt.0)cnt=cnt+1
enddo
rat=cnt/1000000.
x=10/36.
print 1, rat,x
1 format('sim: prob. up = ',f5.3,' analytic = ',f5.3)
end
</pre>
soln - w/ logic and computer
