Question 270096: General Buck is preparing to make his annual budget presentation to the U S senate and is speculating about his chances of getting all or part of his requested budget approved. From his 20 years of experience in making these requests, he has deduced that his chances of getting between 50 and 74 percent of his budget approved are twice as good as those getting between 75 and 99 percent approved, and two and one half times as good as those of getting between 25 and 49 percent approved.Further the general believes that there is no chance of less than 25 percent of his budget being approved.Finally the entire budget has been approved only once during the general's tenure and the general do not expect this pattern to change.What are the probabilities of 0 to 24 percent,25 to 49 percent, 50 to 74 percent, 75 to 99 percent and 100 percent approval, according to the general?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! It looks to me like you have to build this up and then break it down again.
Here's how I see it.
20 years experience means there were 20 annual budgets in total.
His deductions are based on those 20 years and so the percentages must have been derived from that.
I assigned letters to each of the categories.
a = 0 to 24%
b = 25 to 49%
c = 50 to 74%
d = 75 to 99%
e = 100%
From his statements...
From his 20 years of experience in making these requests (TOTAL BUDGET REQUESTS ARE 20), he has deduced that his chances of getting between 50 and 74 percent of his budget approved are twice as good as those getting between 75 and 99 percent approved (THIS MEANS THAT C = 2 * D), and two and one half times as good as those of getting between 25 and 49 percent approved (THIS MEANS THAT C = 2.5 * B).Further the general believes that there is no chance of less than 25 percent of his budget being approved (THIS MEANS THAT A = 0).Finally the entire budget has been approved only once during the general's tenure and the general do not expect this pattern to change (THIS MEANS THAT E = 1).
SUMMARY OF WHAT WE HAVE:
a = 0
c = 2*d
c = 2.5*b
e = 1
total = 20
Since the sum of the parts equals the whole, we get:
a + b + c + d + e = 20
a = 0.
e = 1.
c = c.
Since c = 2*d, this means that d = c/2.
Since c = 2.5*b, this means that b = c/2.5
Our equation becomes
0 + c/2.5 + c + c/2 + 1 = 20
Multiply both sides of this equation by 5 to get:
0 + 2c + 5c + 2.5c + 5 = 100
Combine like terms to get:
9.5c + 5 = 100
Subtract 5 from both sides of this equation to get:
9.5c = 95
Divide both sides of this equation by 9.5 to get:
c = 10
Using the value of c we have obtained, we get:
a = 0
b = c/2.5 = 4
c = 10
d = c/2 = 5
e = 1
Add these all up together and we get 20 = 20 which is good so these values appear to be what we are looking for.
The question was, however:
What are the probabilities of 0 to 24 percent,25 to 49 percent, 50 to 74 percent, 75 to 99 percent and 100 percent approval, according to the general?
The probabilities for each are the occurrences of each divided by the total occurrences.
We get:
p(a) = 0
p(b) = 4/20
p(c) = 10/20
p(d) = 5/20
p(e) = 1/20
total probability is 20/20 = 1 as it should be.
You can reduce the fractions to their simplest terms or make them into decimals at you discretion.
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