In this problem were given that a student takes a multiple choice exam with 10 questions, each of which has four possible selections for the answer. We're also told that the passing grade is 60 or better and we are to suppose that the student was unable to find time to study for the exam and just guesses at each question. So in the first part of the question a we are finding the probability that the students gets at least one question, right? So we have to design this distribution and give the successful ability P and considering that there are four possible selections and only one answer, There is a 25 chance that the student is going to get a question. Right? So that means that the successful ability is 0.25. And since there are 10 questions to be answered in this exam and Is going to be equal to 10. No. So we need to get the probability in the first part of the question. A we need to get the probability that X equals or is greater than one. This is the probability that the students get at least one question, right? So the probability that the students get at least first question right means We will be adding the probabilities. The student got one All the way through until we get to the student got 10 questions. In other words, we would have to get one minus the probability that the student did not get any question. Right? So the best thing to do here is to come up with the probability distribution and to do that, we would need to create a column for X and a column for the probability put of the possible outcomes. So we have zero, one, 23 all the way until we get 10. So it's possible that the student would get uh anything from 0 to 10. So the probability that the students get zero is obtained by the formula. You press the equal button and then the binomial distribution. Then click on the 0 to be the number of questions that were gotten right out of 10 simple questions, Probability the successful ability, 0.25. Mhm. And then we check that we are selecting the probability mass function. Mhm. And we would need to copy that formula all the way down to get their probabilities for the different outcomes. And in this case we see now that we want to work out one minus the probability that X equals zero, which is 0.0 56 And when we worked that out, the probability is 0.944 in the second part of the exercise etc. We are finding the probability that the student passes the exam And we have been told that the past mark is 60 or better for the student to get 60 or better then the number of questions that are correct should be six or more. So we're looking at the probability that X is greater than Or equal to six. And this would have to be the probability The x equal six plus the probability An x equal seven. We keep adding until we get to the probability At X equals 10. If the student gets all the questions right? So the probabilities are given here so we need to work out the sum of all these probabilities. And so you can put a formula that gets the sum of all this. So you say it's equal to yeah. Some sorry, Probability that he gets he or she gets 6789 or 10 that yes zero 01 97 Yes. In part C we're finding the probability that the student receives an A on the exam Uh getting an a means that the student gets 90 or better. So 90 in this case means that the students get nine questions or 10 questions right? So we are simply getting the probability That X is greater than or equal to nine. And that means the probability And x equals nine Plus, the probability that x equals 10. And in this case we need to add those two probabilities and we can improve the formula. Some of the last two grows And that gives us 2.9 six. Mhm. Times 10 to the power of -5. It is a very very small chance that these students is going to get 90 or better next. In part D we are looking at how many questions we would expect the students to get correct and in that case we would have to apply their the meme of this distribution. So the mean of the distribution is given by N times P And in this case N is 10 And the success probability P is 0.25, so That gives us 2.5. So that mean is 2.5 questions. So on average, out of the 10 questions, we expect that the student would get 2.5 of them right out of the guessing. And in part of the question, we're supposed to obtain the standard deviation of the number of questions that the students get gets correct. So the standard deviation is given by the formula, the square root of n times p times one minus p. And in this case it's going to be the square root off 10 Times 0.25. The success probability times One man a 0.25, which is 0.75. And when you work that out, you get the square root of 1.875 and that is equal to 1.36 nine questions Yeah.