In Exercises 6–10, use the following results from tests of an ...

Question

In Exercises 6–10, use the following results from tests of an experiment to test the effectiveness of an experimental vaccine for children (based on data from USA Today). Express all probabilities in decimal form.

Find the probability of randomly selecting 2 subjects without replacement and finding that they both developed flu.

Find the probability of randomly selecting 2 subjects without replacement and finding that they both developed flu.

Accepted Answer

Welcome back, everyone. In a study to test the effectiveness of a new dietary supplement for improving concentration among students, the following data were collected based on exam performance. Find the probability of randomly selecting two students without replacement and finding that they both improved concentration. Round your answer to 5 decial places. Let's label the event as A, which represents that both students improved concentration, and we have to recall that the probability. Of a can be expressed as the ratio of two numbers A and B. We don't know what they are. A is the number of favorable outcomes. And B is the total number of outcomes. So let's focus on this problem. We are going to identify favorable outcomes as Choosing two students out of those who improved concentration. 2 from I see improved concentration. So essentially, we can just define A as the number of ways to choose two students from those who improved concentration. And the total is going to be choosing 2 students from total. Or the total number of students. So now how many of those students improved concentration? Well, we have supplement group and control group. So we have to add 125 plus 120, which gives us a total of 145. And now, how many students do we have total? We have to add all four numbers. Basically 25 + 120 plus those who did not improve concentration, 975, 880. So the total number is going to be 2000. And now we can identify A and B. We're going to use the formula for combinations and choose K, and that's because the order of selection does not matter, and it is without replacement, right? The formula for N choose K isnfactorial divided by K factorial, which is multiplied by N minus K factorial. And now A is the number of favorable outcomes, choosing 2 from those who improved concentration. So those who improved concentration are 145, and now we are applying the formula 145, choose 2. Now, B, our denominator is choosing 2 from total, and the total number is 2000, so that's 2000 choose 2. So the probability that we're interested in is the ratio between A and B, which is 145 choose 2 divided by 2000 choose 2. According to the formula, that would be 145 factorial, divided by 2 factorial. Which is multiplied by 143 factorial. 145 minus 2, right? And then we are dividing by 2000 pictorial. Which is divided by two factorial multiplied by 1,998 pictorial. When we perform the calculations, we're going to get a 0.00522, rounded 2. 5 decimal places and that will be our final answer. Thank you for watching.

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