Asatellite 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 α, what is the probability that the satellite system will function?

Step-by-step explanation:

We will call C the number of components that will function tomorrow. R is the event 'tomorrow rains', and D is the event 'tomorrow is a dry day'. The satellite system will work if .

We can calculate the probability of the event dividing in 2 cases, depending on if it rains tomorrow (R), of if is a dry day (D). According to the theorem of total probability, we have this formula

Since D is the complementary event of R, we have

Note that if we assume that it will rain (or not) tomorrow, then C is a random variable of binomial distribution with parameters n and (or ). To calculate the probability of C being greater than k, we can take probability of the complementary event, in other words, the probability of the event 'C is any j with j between 0 and k-1' and substract it from 1. Therefore

Using this calculations, we can conclude that

I hope it helps you!

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