So 45% of people know someone who is addicted to a drug over than alcohol. So this is a binomial random variable because there are two options you ever addicted to this drug or you're not. So that means we can use this equation to find probabilities because this gives us probabilities of binomial random variables. And I think you should have this running down now for the 1st 1 buying probability of free from a sample of five. So the free pieces of information we need our five, the free and then this 45 up here five is our sample that's N X is gonna be free. That's the number we're looking for. And a probability is 45% Now. We're going to convert this to a decimal, though, so it's going to be point for five. So to find the probability of free that will be equal to the sample. Five choose free times the probability so 0.45 Tioga to the one minus point. Bert multiplied by one minus 10.45 raised to the five minus free power. And this is going to give us 0.276 Now we have a sample of 15 people, and we need to find the probability that seven of them are addicted to the drug. But you need to use the table in the back of the book and appendix B table to it gives you a table that looks something like this. I cannot post pictures directly from the books. I recreated some of it. So you're gonna scroll down to the end equals 15 section. And because our probability is 45% you're gonna be in between the 450.40 and the 0.50 So I wrote out some of the relevant section here. So the fine probability of seven Well, we go to seven. And because it's 70.45 we're just gonna take the average of these two numbers right here. So probability of seven is equal 2.177 plus 0.196 over to. And we're doing this because we're just taking the average of these two, and this gives us 0.1865 Now, these last to find the probability that it's a least seven, So we're still gonna be in this section. So to find the probability of a least seven, you're going to start with seven. So that's here and again. We need both, and you're going to go all the way down until you reach X equals 15. We'll keep going down until you get two X equals 15 and you're gonna add all of these up and divine my chill to find the average. Now, you should get 0.5 or 25 or probability of at least seven. And now, for our last part, find the probability of less than or equal to seven. So hopefully you didn't clear your calculator because if you didn't clear your calculator, all you gotta do is delete this 0.177 and this 0.196 and then same thing. Divide the rest of them by two and then one minus it. However, if he didn't do that, what you're gonna do now is starting again at seven. You're going to go up all the way to X equals zero. So you're gonna take all the cases from zero up to seven for the 70.4 in the 0.5 column Adam all up and divide by two to take the average, just like in the last one, and you should get probability of less than or equal to seven is equal 2.644 Now, if you didn't clear your calculator, just delete the 0.177 in the 0.196 I don't give you the probability of a coup least eight and then one minus it, because the probability of greater than or equal to seven is equal toe one, minus all the other probabilities. Since all probabilities at upto wine, you can save a little bit of time here if you used a calculator and we are done.