91Ó°ÊÓ

A satellite system consists$n$of components and functions on any given day if at least $k$of the $n$components function on that day.

The satellite system may function either tomorrow is rainy or dry.

Whether it will rain or not, the system will function for $i=k$or $k+1$or $…….n$components out of $n$components functions on this day, thus we have for each $i$a total number of combinations equal to $\left(\begin{array}{l}n\\ i\end{array}\right)$.$\left(\begin{array}{l}n\\ i\end{array}\right).$

Let $P(fâˆ\pounds r)$denote the probability that the satellite will function on a rainy day.

$P(fâˆ\pounds r)=\underset{i=k}{\overset{n}{∑}}\left(\begin{array}{l}n\\ i\end{array}\right)p_1^i{\left(1-p_1\right)}^{n-i}$

Let $P(fâˆ\pounds d)$denote the probability that the satellite will function on a dry day.

$P(fâˆ\pounds d)=\underset{i=k}{\overset{n}{∑}}\left(\begin{array}{l}n\\ i\end{array}\right)p_2^{\text{'}}{\left(1-p_2\right)}^{n-i}$

Calculate the probability that the satellite system will function.

$P\left(f\right)=P(fâˆ\pounds r)P\left(r\right)+P(fâˆ\pounds d)P\left(d\right)$

$=\underset{i=k}{\overset{n}{∑}}\left(\begin{array}{l}n\\ i\end{array}\right)p_1^i{\left(1-p_1\right)}^{n-i}\mathrm{Î\pm }+\underset{i=k}{\overset{n}{∑}}\left(\begin{array}{l}n\\ i\end{array}\right)p_2^i{\left(1-p_2\right)}^{n-i}(1-\mathrm{Î\pm })$

We get,

$=\mathrm{Î\pm }\underset{i=k}{\overset{n}{∑}}\left(\begin{array}{l}n\\ i\end{array}\right)p_1^i{\left(1-p_1\right)}^{n-i}+(1-\mathrm{Î\pm })\underset{i=k}{\overset{n}{∑}}\left(\begin{array}{l}n\\ i\end{array}\right)p_2^i{\left(1-p_2\right)}^{n-i}$

The probability that the satellite system will function.$=\mathrm{Î\pm }\underset{i=k}{\overset{n}{∑}}\left(\begin{array}{l}n\\ i\end{array}\right)p_1^i{\left(1-p_1\right)}^{n-i}+(1-\mathrm{Î\pm })\underset{i=k}{\overset{n}{∑}}\left(\begin{array}{l}n\\ i\end{array}\right)p_2^i{\left(1-p_2\right)}^{n-i}$