For this problem. We're told that 2% of two million high school students would take the S A T every year, requires special accommodations and then were asked to consider a random sample of 25 students who have recently taken the test. And we're asking probabilities so weakened we can define X as the number of successes being the number of students who required special accommodation and because we're sampling without replacement from the population. The samples are not strictly independent, but because 25 is far less than 5% of the total population, X can be pretty reliably estimated as a binomial random variable based on 25 size of 25 and the probability of success of 0.2 part A were asked, What is the probability that exactly one required special accommodation? What is the probability that we have exactly one success? So that's the probability mass function for the binomial, and this comes out to approximately zero point 308 now for B were asked, what is the probability that at least one received special accommodation? So what is the probability that we have at least one success we can rewrite this as one minus the probability of getting exactly zero successes, and this comes out to approximately zero point 397 or C. We're looking for the probability of at least two successes, so this could be rewritten as one minus probability. That X is less than or equal to one, and this equals one minus 0.911 which equals 0.89 for a party, you were asked, what is the probability that the number of successes is within two standard deviations of the mean number of successes or the expected number of successes? So to calculate this, we must calculate the mean number of successes where this distribution and the standard deviation So for a binomial random variable, the mean or expected value for the number of successes is he going to end terms? P, which comes out to you 1/2 and the standard deviation of a binomial random variable is given by the square root of N times. P Times Q, which comes out to 0.7, so within two standard deviations of the expected value. We're within this range, so the mean value is 0.5 and so plus or minus two times 0.7, which is the standard deviation, and that gives us a range from minus 0.9 to 1.9. So that means we're looking for the probability of a certain number of successes. So we for outcomes we can only have integer numbers of students, and that can't be negative. So this would be replaced by zero Can't be any lesson zero, and the number two would be more than two standard deviations greater than the expected number. So therefore, we must look for outcomes that are less than or equal to one. So we're looking for the probability that the number of successes is between zero and one, which is the same as saying we're looking for the probability of at most one success, which comes out to 0.911 and finally for party. It's explained to us that students who receive special accommodation are allowed 4.5 hours for the S A T, and students who do not are allowed three hours to rate the S A T. So we're asked, what do we expect? The average time allowed for the 25 students to be, so we can begin by writing a function that defines the average time spent by the 25 students. So for a student who is given special accommodation to get 4.5 hours, so we have 4.5 hours times number of students who get the special accommodation and then the students who don't get three hours and the number of them will be 25 minus the number of successes. And to make it an average, we must divide by the number of students, divide by 25. So this function defines the average number of hours that the students are given for the S A. T out of a sample of 25 and this can be simplified as 75 plus 1.5 x over 25. So for the question, we're asked, what is the number of hours allocated to the students on average, that we expect. So what is their expected number for the average number of hours allocated to the students? So what is asking for us? The expected value of the function h of X. So we can say that is equal to the expected value of 75 plus 1.5 x over 25. And because of the linearity of expectation, we can factory with the 1/25 and rewrite the rest of it like this. Now we have 1/25 so the expected value of a constant is just that constant. So we have 75 plus 1.5 times the expected value for X. And we've already calculated that previously, as 0.5. So this comes out to 3.3 So are we expect that the average number of hours allocated to the students to the 25 students writing S A T will be 3.3 hours.