Question

A satellite system consists of$n$ components and functions on any given day if at least $k$ of the n components function on that day. On a rainy day, each of the components independently functions with probability $p1,$ whereas, on a dry day, each independently functions with probability $p2$. If the probability of rain tomorrow is what is the probability that the satellite system will function?

Short answer

The probability that the satellite system will function$=\alpha \sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_1^i{\left(1-p_1\right)}^{n-i}+(1-\alpha )\sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_2^i{\left(1-p_2\right)}^{n-i}$

Step-by-step solution

Given Information

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 $\dots \dots .n$components out of $n$components functions on this day, thus we have for each $i$a total number of combinations equal to $\left(\begin{array}{l}n\\ i\end{array}\right)$.$\left(\begin{array}{l}n\\ i\end{array}\right).$

Solution of the Problem

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

$P(f\mid r)=\sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_1^i{\left(1-p_1\right)}^{n-i}$

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

$P(f\mid d)=\sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_2^{\text{'}}{\left(1-p_2\right)}^{n-i}$

Computation of the Probability

Calculate the probability that the satellite system will function.

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

$=\sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_1^i{\left(1-p_1\right)}^{n-i}\alpha +\sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_2^i{\left(1-p_2\right)}^{n-i}(1-\alpha )$

We get,

$=\alpha \sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_1^i{\left(1-p_1\right)}^{n-i}+(1-\alpha )\sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_2^i{\left(1-p_2\right)}^{n-i}$

Final Answer

The probability that the satellite system will function.$=\alpha \sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_1^i{\left(1-p_1\right)}^{n-i}+(1-\alpha )\sum _{i=k}^n\left(\begin{array}{l}n\\ i\end{array}\right)p_2^i{\left(1-p_2\right)}^{n-i}$