A club with 50 college students is doing volunteer work this semester. Each student is volunteering at one of four locations. Here is a summary.Location Number of studentsNursing home 14Soup kitchen 15Hospital 8Pet shelter 13Three students from the club are selected at random, one at a time without replacement. What is the probability that none of the three students volunteer at the hospital?Do not round your intermediate computations. Round your final answer to three decimal places.(If necessary, consult a list of formulas.)
Question
A club with 50 college students is doing volunteer work this semester. Each student is volunteering at one of four locations. Here is a summary.Location Number of studentsNursing home 14Soup kitchen 15Hospital 8Pet shelter 13Three students from the club are selected at random, one at a time without replacement. What is the probability that none of the three students volunteer at the hospital?Do not round your intermediate computations. Round your final answer to three decimal places.(If necessary, consult a list of formulas.)
Solution
To find the probability that none of the three students volunteer at the hospital, we need to calculate the probability of each student not volunteering at the hospital and then multiply those probabilities together.
First, let's calculate the probability that the first student does not volunteer at the hospital. Out of the 50 students, 8 are volunteering at the hospital, so the probability that the first student does not volunteer at the hospital is (50 - 8) / 50.
Next, let's calculate the probability that the second student, who is selected without replacement, does not volunteer at the hospital. After the first student is selected, there are now 49 students left, and 7 of them are volunteering at the hospital. So the probability that the second student does not volunteer at the hospital is (49 - 7) / 49.
Finally, let's calculate the probability that the third student, also selected without replacement, does not volunteer at the hospital. After the first two students are selected, there are now 48 students left, and 6 of them are volunteering at the hospital. So the probability that the third student does not volunteer at the hospital is (48 - 6) / 48.
To find the overall probability that none of the three students volunteer at the hospital, we multiply these probabilities together:
[(50 - 8) / 50] * [(49 - 7) / 49] * [(48 - 6) / 48] = 0.764
Therefore, the probability that none of the three students volunteer at the hospital is approximately 0.764, rounded to three decimal places.
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