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Probability Help

Posted on 2016-07-21
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Last Modified: 2016-07-26
Can you please  help me figure this out? Thanks.

Suppose managers at a company wish to meet the increasing demand for color photocopies and to have more reliable service. Based on historical data, the chance that a black-and-white copier will be down for repairs is 0.12. The color copiers are more of a problem and are down 23% of the time. As  a goal,  they would like to have at least a 99.90% chance of being able to furnish a black-and-white copy or a color copy on demand. They also wish to purchase only four copiers. They have asked for advice regarding the mix of black-and-white and color copiers. Supply them with advice on how to purchase the greatest number of color copiers while also meeting their goal regarding the furnishing of a copy on demand.  Provide calculation and reasons to support that advice. (Assume that the color copier can also be used to make a black-and- white copy.)

The manager should purchase
a. 2
b 1
c. 4
d. 0
e. 3

black-and-white copier(s)  and
a. 1
b. 0
c. 4
d. 3
e. 2

color copier(s).  If they do this, they will have a  ____  % chance of being able to furnish at least a black-and-white copy or a color copy on demand.
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Question by:mustish1
3 Comments
 
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Accepted Solution

by:
Petr J earned 250 total points
Comment Utility
Probability of all N photocopiers being broke at a time is p^N (p is probability of 1 photocopier being broken).
This brings us to quite easy formula:
p_BW^c_BW*p_C^c_C >= 1-0,999
where
p_BW is probability black-and-white is broken
c_BW is count of black-and-white photocopiers
and the same for color (C) photocopiers.

Since BW photocopiers are more reliable and there is no demand for C photocopiers we can simplify the formula to:
p_BW^c_BW >= 0,001

After putting logarithm on both sides we come up with:
c_BW * log(p_BW) <= log(0,001)
c_BW * log(0,12) >= log(0,001)
c_BW >= log(0,001) / log(0,12)
c_BW >=3,26

The solution is to buy 4 BW photocopiers (this is minimal count of photocopiers).

3 BW and 1 C photocopier would meet the criteria as well:
0,12^3*0,23=0,0004<=0,001

2 BW and 2 C photocopier would meet the criteria as well:
0,12^2*0,23^2=0,0008<=0,001

1 BW and 3 C photocopier would not meet the criteria:
0,12^1*0,23^3=0,00146004>0,001
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Assisted Solution

by:d-glitch
d-glitch earned 250 total points
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This question is unrealistic and ambiguous, but you still have to have to start.
Pick any scenario and analyze it.
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Author Closing Comment

by:mustish1
Comment Utility
Thank You.
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