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

Question

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

25. Best President In a sample of 1500 adult U.S. citizens, 270 said that Barack Obama was the best president in U.S. history. Two adult U.S. citizens are selected at random.

(Adapted from YouGov)

a. Find the probability that both adult U.S. citizens say that Barack Obama was the best president in U.S. history.

Accepted Answer

Welcome back, everyone. Out of a group of 1200 employees, 156 said they preferred remote work over in-office work. If two employees are randomly selected without replacement, what is the probability that both prefer remote work? A says it's about 0.0121, B about 0.0139. C about 0.0168. And about 0.0297. No Let's first define our events, OK? So let's let A be the probability or B the event, OK? Let A be the event. That the first employee. Prefer the remote work. And now, let's let B, OK, it's basically gonna be the same statement but just a change for our second employee. So now let's let B. OK. Let be be the event that the 2nd employee prefers remote work, OK? 2nd employee. Now, how can we find the probability that both prefer remote work? Well, let's think about our two events, OK? Now, we're selecting individuals from a finite group without replacement and because of that, the outcomes of each selection influence one another. So for example, in our group of, let's say this is our group of 1200 employees, OK, then from our group of 1200 employees. Then in our first scenario. 156 preferred remote work. OK, so let's say this part that's shaded red was the one that preferred remote work. And if we choose one and they actually are, or they actually do prefer remote work, then in the second scenario, no, we have a smaller pool to choose from, OK? Because now if we think about it. It's one less success which is one less employee so no instead of the whole 1200, OK, instead of 1200 employees, no, we're gonna have 1 less or 1,199 employees to choose from. And from those 1,199 employees, no, 155 would prefer remote work, OK? Because remember, we've taken out one that already preferred remote work. So no, In that case, OK. This change in the composition of the group means that the probability for the 2nd event is affected by the outcome of the 1st event. Therefore, the two events selecting a remote preferring employee first and then selecting another remote preferring employee are dependent events. And recall that for dependent events, the probability that both events happen, the probability of A and B is going to be equal to the probability of a multiplied by the probability of B given A, that is the probability of event B occurring given that event A has already occurred. No, from our diagram first, this one would would represent the probability of A, OK. And this one would represent the probability of B. So now if we think about it. This is gonna be equal to the number of successes 156 divided by the total outcomes 1200 multiplied by the number of successes in B given A, that is 155 divided by the number of successes 1,199. I know when we multiply both of these, it's approximately equal to 0.0168. Thus, the probability that both employees prefer remote work is about 0.0168. If we look at our answer choices, that means C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

��app

Key Concepts

Multiplication Rule of Probability

Probability Calculation

Independent Events

Watch next