In this question, we're told that 20% of individuals have an adverse reaction to a particular drug, so we can see that probability is 0.2 and the drug is administered two individuals, one after another until the first adverse reaction. So we're looking at the number of trials until we have exactly one success, and we're as to define an appropriate random variable and use its distribution to answer the questions and the questions. A through E all pertained to how many trials must occur before we have one adverse reaction. So we can define our random variable X as the number of drug administrations until the first adverse reaction. So if you're thinking this is a negative binomial distribution, that is right, it's also a geometric random variable. And that's just a special case of the negative binomial. When R is equal to one so far, a geometric distribution, the probability mass function is simply one minus p to the exponents, X minus one times p and that is for X is a positive integer for part. A were asked the probability that when the experiment terminates that four individuals have not had adverse reactions, so that means we have the first four individuals do not have a reaction, and then the fifth one has reaction, and that's when it terminates. So this can only be the probability that we have five trials. That's four no reactions, followed by one adverse reaction. So it's 0.8, which is one minus p, and this probability comes out to 0.82 in part B. We're looking for the probability that the drug is administered to exactly five individuals. So this is the same question. Is part a just worded in a different way. If you have four non adverse reactions, followed by one adverse reaction that is the same as having the drug administered to a total of five people for part C, we're asked the probability that at most, four individuals do not have an adverse reaction. That is the same as the probability that at most, five drug administrations take place before it ends. So if we have at most five people being administered the drug, we know that at most, four have had no reaction. So this is the summation from X equals 1 to 5 of 0.8 to the exponents X minus one time 0.2 and this probability is 0.672 for part D were asked how many individuals we would expect do not have an adverse reaction And how many individuals would you expect to be given the drug? So the number that we expect to be given the drug will be the number without an adverse reaction, plus one. So the expected value for X This is the expected number of drug administrations in total for a geometric random variable is given by one overpay. So we expect five people to have been administered the drug, and therefore we expect four people to have not had an adverse reaction, and for party were asked for the probability that the number of admit of individuals given the drug is within one standard deviation of what we expect so mathematically, this could be written as what we expect is the mean, so minus one standard deviation. We're saying, What is the probability that X is within one standard deviation of the mean like that? And so we already know the mean That's from part D. We just have to calculate the standard deviation So the variance on X for a geometric random variable is one minus p over P squared, which comes out to 20. So if we take the square root of that, we get the standard deviation. That's 4.47 So we have that. We're looking for the probability that five minus 4.47 is less than or equal to x. Just listen or equal to five plus 4.47 I recall that X must be an integer. So this comes out to the probability that one is less than X lessner equaled X, which is less than or equal to nine. So the number of trials What is the probability that the number of trials is between one and nine inclusive, more than nine would be more than one standard deviation away from the mean and less than one would be more than one standard deviation away from the mean. Of course, we can't actually have less than one, because this is only defined for excess one or greater. You have to have at least one trial so we can write this as a summation for X equals one 3 to 9 for our probability mass function. And this calculation comes out to 0.866