In Exercises 49-53, use counting principles to find the probab...

Question

In Exercises 49-53, use counting principles to find the probability.

52. A class of 40 students takes a statistics exam. The results are shown in the table at the left. Three students are selected at random. What is the probability that

b. all three students received a C or better?

Table showing letter grades and the number of students: A-8, B-10, C-12, D-6, F-4.

Accepted Answer

Hello everyone. Let's take a look at this question together. A total of 40 employees completed a workplace safety course. Their ratings are summarized below, and here we have a table of the different ratings and the number of employees that gave each rating. If 3 employees are selected at random without replacement, what is the probability that all 3 received a rating of satisfactory or better? Is it answer choice A, 0.35? Answer choice B, 0.05? Answer choice C 0.50, or answer choice D 0.59. So in order to solve this question, we have to determine the probability that all three employees received a rating of satisfactory or better if the three employees are selected at random without replacement. And so the first step in solving this problem is to identify the ratings of satisfactory or better. Which these ratings include The excellent rating that had 10 employees, the good rating that had 14 employees, and the satisfactory rating that had 8 employees. Therefore, the total favorable employees is 10 + 14 + 8, which is equal to 32. And then using that data, we can use the formula for combinations, where the number of ways to choose our amount of people from a group of N is using the formula N over R, which is equal to N factorial divided by R factorial multiplied by N minus R factorial. So the total number of ways to choose any three employees from 40. Using the formula for combinations where we plug in 40 for N and 3 for R, simplifying the expression, we get 9880 as the total number of ways to choose any three employees from 40. And then we calculate the same for the number of ways to choose 3 employees from the 32 who rated satisfactory or better. Using the formula for combinations where we plug in 32 for N and 3 for R and simplifying, we get 4960 as the number of ways to choose 3 employees from the 32 who rated satisfactory or better, and then using both of those values, we can compute the probability. Which is the probability that all three rated satisfactory or better, and is given as the favorable outcomes out of the total amount of outcomes, which when substituting the values, we get 4960 divided by 9880, so the probability that all three rated satisfactory or better is approximately 0.50. Which, looking at our answer choices, we can identify is answer choice C, so answer choice C is correct, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

In Exercises 49-53, use counting principles to find the probability.
52. A class of 40 students takes a statistics exam. The results are shown in the table at the left. Three students are selected at random. What is the probability that
b. all three students received a C or better?

Key Concepts

Counting Principles

Probability

Combinations

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