In this question, we're talking about K out of end systems and that each of the end components in each system have a probability of working of 0.96 so we could define that as a success for a given component. If it works, that is a success, and we can consider a K out of N system as looking at end. Samples in each of these n trials can either have a success or a failure. So basically a K out of an system can be viewed. As with a number of successes can be viewed as the number of components in the kit of end. That air working can be viewed as binomial random variable were there en trials and their probability of success is 0.96 So for part A were given a three component system and arrest. What is the probability that exactly two components function? So here we can say the number of components functioning is the binomial random variable based on three trials and a probability of success of 0.96 and we want the probability that exactly two of the components successful successfully function, which comes out to a probability of approximately zero point 111 now for B were asked, What is the probability that at two other three system works, two out of three system means that we must have at least two components working? So what is the probability that the number of successes is at least two? It's gonna be written as one minus probability of, at most one success, which comes out to about zero point 995 now. CS. What is the probability of a three out of five system working? So if the number of successes here, a number of successful components is a binomial random variable based on five trials and the probability of success of 0.96 so we need at least three of the components to work. 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. And this probability probability of this system working is approximately 0.999 four he is. What is the probability of a four out of five system working so again we're using this seem random variable at this time We need at least four successes, which comes out to about 0.985 and finally for party were asked, What does the component probability need to be so that four out of five system will function with probability at least 0.9999? So we're talking about a number of successes is a binomial random variable and there's five trials. It's, ah, five component system, and we'll give it probability successes p. So we're trying to solve for P. And we want the probability that it works, which is the probability that, at least or components function to be greater than or equal to 0.9999 We could also write this the probability that exactly four components, plus the probability that exactly five components work is greater than or equal to zero point 9999 Now, if we use the equation for the probability mass function for a binomial random variable, we can make this equation look like this. So it's so the first term is five. Choose for times P to the exponents four times one minus p to the experiment one plus five choose five times, pleaded exponents five. And this comes out to minus four times p to export and five plus five times P. Diddy Explode and four. And if we subtract 0.9999 from both sides and really what we could do? Here's to find the minimum value for P. We can just equate this to zero, and so we have to solve four P in this equation. So I used software to sell for this and give me P is equal to about 0.9969 So in order for a four out of five system toe work, with probability at least 0.9999 we need P to be at least that reliable. So we need P to be greater than or equal to zero point 9969 where p is the component reliability.