In Exercises 29 and 30, find the probabilities and indicate wh...

Question

In Exercises 29 and 30, find the probabilities and indicate when the “5% guideline for cumbersome calculations” is used.

Medical Helicopters In a study of helicopter usage and patient survival, results were obtained from 47,637 patients transported by helicopter and 111,874 patients transported by ground (based on data from “Association Between Helicopter vs Ground Emergency Medical Services and Survival for Adults with Major Trauma,” by Galvagno et al., Journal of the American Medical Association, Vol. 307, No. 15).

b. If 5 of the subjects in the study are randomly selected without replacement, what is the probability that all of them were transported by helicopter?

Accepted Answer

Welcome back everyone. In a study of a new flu vaccine, 8500 people were vaccinated and 17,500 were not vaccinated. If 4 people are randomly selected without replacement, what is the probability that all of them were vaccinated? Round your answer to 4 decimal places. We're looking for the probability of an event A where A represents selection of 4 people who were vaccinated. And we want to recall that probability is simply a ratio between two numbers A and B. A is the number of favorable outcomes. And B is the total number. Of outcomes. So let's identify A and B. We're going to begin with B, which is our denominator, and we're going to choose. The formula for combinations, which is N choose K. Now why are we using combinations? Well, we are performing selection without replacement in the order of selection does not matter. Let's recall the formula. It says that we're taking N factorial, dividing by K factorial, which is multiplied by N minus K factorial. Alternatively, we can just use a calculator to evaluate it. Now, what we're going to do is simply understand that the total number of people is going to be 8500 plus 17,500. The reason why this this number is important is because we want to find The total sample size, which consists of those who were vaccinated and were not vaccinated. When we add these together, we get 26,000 people. So our denominator simply represents the number of ways to choose 4 people out of 26,000 people, meaning B is equal to 26,000. That's the total number of people. Cho K and K is 4 because we're choosing only 4 of them. And our numerator A. Is the number of favorable outcomes, so we want to choose 4 people from those who were vaccinated, meaning now our N becomes 8000. 500 So we get 8500, choose 4. Simply speaking, we want to choose 4 out of those who were vaccinated. And we can now evaluate. The probability, which is a divided by B or simply 8500 choose 4 divided by 26,000 choose 4. We can use a calculator to evaluate the answer immediately, or we can use the formula, right? So for our numerator. We get 8500 factorial. Divided by K factorial, which is 4 factorial, multiplied by 8500 minus 4 factorial. And this is divided by 26,000 factorial. Divided by or factorial, multiplied by 26,000 minus 4 factorial. And when we perform the calculations, we're going to get 0.0114 rounded to 4 decimal places. That will be our final answer, and thank you for watching.

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Key Concepts

Probability

Sampling Without Replacement

5% Guideline for Cumbersome Calculations

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