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Problem 82: It has been claimed that in 60% of all solar-heat installtions the utility bill is reduced by at leasr one-third.

Problem 82: It has been claimed that in 60% of all solar-heat installtions the utility bill is reduced by at leasr one-third. Accordingly, what are the probablities that the utility bill will be reduced by at least one-third in


a . four of five instalaltions; b. at least four of five instalaltions?

Solution:


a. Given that $x=4,n=5,$ and $p=0.6$
Binomial distribution given by
$b(x;n,p)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}(1-p)^{n-x}$
$b(4;5,0.6)=\left(\begin{array}{c}5\\ 4\end{array}\right){0.6}^{4}(1-0.6)^{5-4}$
$=^{5}C_{4}(0.6)^{4}(0.4)^{1} = 5(0.6)^{4}(0.4)^{1} = 0.2592 $. a. Given that $x=5,n=5,$ and $p=0.6$
Binomial distribution given by
$b(x;n,p)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}(1-p)^{n-x}$
$b(5;5,0.6)=\left(\begin{array}{c}5\\ 5\end{array}\right){0.6}^{5}(1-0.6)^{5-5}$
$=^{5}C_{5}(0.6)^{5}(0.4)^{0} = 1(0.6)^{5} = 0.07776 $.
$b(5;4,0.6) + b(5;5,0.6) = 0.2592 + 0.07776 = 0.33696$.

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