Using the Multiplication Rule In Exercises 19-32, use the Mult...

Question

Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.

26. Worst President In a sample of 1500 adult U.S. citizens, 690 said that Donald Trump was the worst president in U.S. history. Three adult U.S. citizens are selected at random.

(Adapted from YouGov)

c. Find the probability that at most two of the three adult U.S. citizens say that Donald Trump was the worst president in U.S. history.

Accepted Answer

All right, hello, everyone. So this question says, in a survey of 1800 college students, 720 reported that they prefer online classes over in-person classes. If 3 students are chosen at random, what is the probability that at most 2 of them prefer online classes? And here we have 4 different answer choices labeled A through D. So the first thing I want to make note of here is how different the sample size is relative to the population. And what I mean by that is that the sample size of 3 is remarkably smaller than the population size of 1800 total students. The reason this is worth bringing up is because removing 1 or 2 students from such a large group has a negligible effect on overall probabilities. For example, the probability that the first student selected prefers online classes is simply equal to 720 divided by 1800, which gives you 0.4. The probability that the 2nd student prefers online classes. Given that one online preferring student has already been chosen, changes only slightly. So instead, that's 719 divided by 1,799. Which is approximately 0.3997. Now, because the difference between probabilities with and without replacement is so small, we can assume independence for practical reasons. And because of that, we can use the multiplication rule. Recall that the multiplication rule for independent events says that the probability of a sequence of independent outcomes is the product of their individual probabilities. So from here you would break down each possible scenario for the 3 students. So case one. To start off with would simply be that no student prefers. Online classes And so to find the probability that zero of the 3 students prefer online classes, you would multiply the individual probabilities. Keep in mind that if 1800 students. Or if there are 1800 students total, and 720 of them prefer online classes, that means that 180 of them do not prefer online classes. So The probability that all three of the chosen students do not prefer online classes is equal to 1080. Divided by 1800. Cube, since it would be the same chances for all of them. And so this gives you approximately 0.216. Now let's move on to the second scenario. Which in this case is that only one student prefers. Online classes. So, the probability that only one student prefers online classes. Would be equal to 720. Divided by 1800. And then multiplied by. 1080 divided by 1800 squared. But keep in mind that this can happen in 3 different ways, because the 1st, the 2nd, or the 3rd student prefers online. Or could prefer online classes, so To this expression, you would also multiply it by 3, to represent the 3 possible ways that this can be the case. So, this gives you 0.432. And now for case 3, which is that 2 of the 3 students prefer. Online classes. So here That would be 720 divided by 1800 squared. Multiplied by 1080 divided by 1800. And multiplied by 3. Because this can also happen in 3 different ways. You can have the 1st 2 students prefer online classes, the 1st and the 3rd student, or the 2nd and the 3rd student. So, the probability of two students preferring online classes is equal to 0.288. So, to find the total probability that at most 2 students prefer online classes, you would add. The individual probabilities that you just calculated. So the total probability. Is equal to the sum of 0.216. 0.432. And 0.288, which gives you 0.936. So from there you have your final answer. But very quickly I want to show you a much quicker method. Just to give you an alternative approach to keep in mind. Earlier we calculated the probability. Or what you could do, rather, is calculate the probability that all 3 students chosen prefer online classes. So You would say, or you would calculate 720. Divided by 1800 cubed. Which would give you 0.064. So to find the probability that at most 2 students prefer online classes, you would subtract. The probability that all three prefer online classes from 1. So that's one subtracted by 0.064, which also gives you 0.936. And there you have it. So if I scroll up here. The correct answer is 0.936, which corresponds to option C in the multiple choice. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

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Key Concepts

Multiplication Rule of Probability

Binomial Distribution

At Most Probability

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