```Question 142057
In the Ardmore Hotel, 20 percent of the customers pay by American Express credit card.
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Comment: This is a binomial problem with mean = np = 20*0.2 = 4 and
standard deviation sqrt(npq) = sqrt(20*0.2*0.8) = sqrt(3.2) = 1.7889
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P(pay by AE) = 0.2 ; P(do not pay by AE)= 0.8
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(a) Of the next 10 customers, what is the probability that none pay by American Express?
p(none in 10) = 0.8^10 = 0.10737...
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(b) At least two?
Comment: "at least" is a key word in binomial problems.  On this problem
you have to understand the following:
P(none pay) + P(one pays) + P(at least two pay) = 1
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P(at least two) = 1 - [P(none) + P(one pay)
= 1 - [0.10737 + 20C1(0.2)^1*(0.8)^19] = 1 - [0.10737+0.05765] = 0.8350
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(c) Fewer than three?
= P(one) + P(two) = 1- 0.8350 = 0.1650
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(d) What is the expected number who pay by American Express?
mean = np = 20*0.2 = 4
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(e) Find the Standard deviation.
sqrt(npq) = sqrt(4* 0.8) = sqrt(3.2) = sqrt(1.6*2) = 0.8sqrt(2)
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Cheers,
Stan H.
```