A hospital has three emergency generators for use in case of a power failure. Ea
ID: 3241526 • Letter: A
Question
A hospital has three emergency generators for use in case of a power failure. Each generator functions independently, and the manufacturer claims that the probability that each generator will function properly during a power failure is 0.95.
1.) List all of the possible outcomes in the sample space, S, and find the probability associated with each outcome.
2.) Define the random variable X = the number of generators that fail. Calculate the value of X for each outcome in the sample space and give the probability mass function for X. Use correct notation.
3.) Find the mean and variance of X.
4.) Suppose that a power failure occurs and all three generators fail. Do you think that there would be reason to doubt the manufacturer‘s claim? Use probability to justify your answer.
Explanation / Answer
Answer:
A hospital has three emergency generators for use in case of a power failure. Each generator functions independently, and the manufacturer claims that the probability that each generator will function properly during a power failure is 0.95.
P( function properly)=0.95
P( failure)= 1-0.95=0.05
SAMPLE SPACE
Probability
WWW
0.857375
WWF
0.045125
WFW
0.045125
WFF
0.002375
FWW
0.045125
FWF
0.002375
FFW
0.002375
FFF
0.000125
2.) Define the random variable X = the number of generators that fail. Calculate the value of X for each outcome in the sample space and give the probability mass function for X. Use correct notation.
x
p
0
0.857375
1
0.135375
2
0.007125
3
0.000125
3.) Find the mean and variance of X.
x
p(x)
x*p(x)
(x-mean)^2*p(x)
0
0.857375
0
0.019291
1
0.135375
0.135375
0.097808
2
0.007125
0.01425
0.024385
3
0.000125
0.000375
0.001015
Total
1.000
0.15
0.1425
Mean =0.15
Variance =0.1425
4.) Suppose that a power failure occurs and all three generators fail. Do you think that there would be reason to doubt the manufacturer‘s claim? Use probability to justify your answer.
Probability of power failure occurs and all three generators fail =0.000125 which is very low.
There is no reason to doubt the manufacturer‘s claim.
SAMPLE SPACE
Probability
WWW
0.857375
WWF
0.045125
WFW
0.045125
WFF
0.002375
FWW
0.045125
FWF
0.002375
FFW
0.002375
FFF
0.000125
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