Remember
Simpler probability problem?!
An urn contains 3 black, 12 white, and 48 gray marbles. They are to be drawn one at a time until all of two of the colors has been exhausted.
(I) What is the probability that the color of the first draw is the same as:
(a) The first color exhausted?
(b) The second color exhausted?
(II) What is the probability that the color of the second draw is the same as:
(a) The first color exhausted?
(b) The second color exhausted?
*** Thanks go to
Jer for inspiring this puzzle.
The keeping track and development of all the possible states seemed daunting so I figured simulation was the way to go:
Ia=0; Ib=0; IIa=0; IIb=0;
for tr=1:10000000
nos=[3 12 48];
r=randi(sum(nos));
pick1=1+sum((r>nos(1))+(r>sum(nos(1: 2))));
nos(pick1)=nos(pick1)-1;
r=randi(sum(nos));
pick2=1+sum((r>nos(1))+(r>sum(nos(1: 2))));
nos(pick2)=nos(pick2)-1;
exhaust1=0;exhaust2=0;
pickNo=3;
while exhaust2==0
r=randi(sum(nos));
pick=1+sum((r>nos(1))+(r>sum(nos(1: 2))));
nos(pick)=nos(pick)-1;
stage=sum(nos==0);
if stage==1
if exhaust1==0
exhaust1=pick;
end
elseif stage==2
if exhaust2==0
exhaust2=pick;
end
end
pickNo=pickNo+1;
end
if pick1==exhaust1
Ia=Ia+1;
end
if pick1==exhaust2
Ib=Ib+1;
end
if pick2==exhaust1
IIa=IIa+1;
end
if pick2==exhaust2
IIb=IIb+1;
end
end
disp([Ia Ib IIa IIb]/tr)
>> facileProbabilitysimulation
0.0959741 0.2910917 0.096009 0.29085
These are the statistics for Ia, Ib, IIa and IIb respectively. The results for II look statistically the same as for I. When you think about it, this is to be expected, as the probability of getting color A in the first draw and B in the second is the same as B in the first and A in the second; and the state of the system is the same after either of these pairs of draws.
For example, the probability of drawing a black marble in the first draw and a gray marble in the second is 3/63 * 48/62. That of a gray followed by a black is 48/63 * 3/62. Each of the two products is the same.
So the (a) probability is about 0.096 and of (b) is about .291.
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Posted by Charlie
on 2022-10-09 10:00:02 |
simulation
