A satellite system consists of n components and functions on...

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 $p_{1},$ whereas on a dry day they each independently function with probability $p_{2} .$ If the probability of rain tomorrow is $\alpha,$ what is the probability that the satellite system will function?

Key Concepts

Binomial Distribution

This concept is essential when dealing with independent trials that have two outcomes – success or failure. In the context of the problem, each component functions (success) with a certain probability and fails with the complementary probability. The binomial distribution is used to calculate the probability of achieving at least a specified number of successes (k out of n), by summing the relevant probabilities across all possible counts meeting the threshold.

Conditional Probability and the Law of Total Probability

The problem involves different scenarios based on weather conditions, each with its own probability of component success. Conditional probability is applied to assess the likelihood of the satellite functioning given a specific weather (rainy or dry day). The law of total probability is then used to combine these scenarios into an overall probability by weighing the conditional probabilities with the probability of each weather condition occurring.

Independent Component Functioning

In the problem, each component operates independently from the others, meaning the functioning or failure of one does not influence the performance of the others. This assumption of independence allows the use of binomial probabilities and greatly simplifies the calculation of the likelihood that a minimum number of components are working by treating each component's status as an independent event.

k-out-of-n Systems

This is a reliability structure where a system is deemed functional if at least k out of n components are operational. This concept is vital in analyzing the overall performance and reliability of systems. It involves calculating the cumulative probability of having at least a threshold number of functioning components, reflecting the system’s ability to operate despite some component failures.

Transcript

00:01 In this question, we are talking about k out of n systems and that each of the n components in each system have a probability of working of 0 .96.

00:13 So we could define that as a success.

00:17 For a given component, if it works, that is a success.

00:21 And we can consider a k out of n system as looking at n samples.

00:28 And each of these n trials can either have a success or a.

00:32 A failure.

00:33 So basically a k out of n system can be viewed as, well the number of successes can be viewed as the number of components in the k out of n that are working can be viewed as a binomial random variable where there are n trials and our probability of success is 0 .96.

01:09 So for part a we're given a three component system and we're asked what is the probability that exactly two components function.

01:19 So here we can say the number of components functioning is a binomial random variable based on three trials and a probability of success of 0 .96.

01:34 And we want the probability that exactly two of the components successfully function, which comes out to a probability of approximately 0 .111.

02:04 Now for b we're asked what is the probability that a two of a two out of three system works.

02:09 So a two out of three system means that we must have at least two components working.

02:16 So what is the probability that the number of successes is at least two? this can be written as one minus probability of at most one success, which comes out to about 0 .995.

02:58 Now see what is the probability of a three out of five system working? so if the number of successes here, the number of successful components is a binomial random variable based on five trials and the probability of success of 0 .96.

03:21 So we need at least three of the components to work.

03:26 So what is the probability that x is at least three? it's equal to one minus the probability that x is at most two.

03:51 And this probability, probability of this system working is approximately 0 .994...

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