Question

a club with 55 college students is doing volunteer work this semester. each student is volunteering at one of four locations. here is a summary.

location	number of students
soup kitchen	9
library	13
pet shelter	18

three 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 library?
do not round your intermediate computations. round your final answer to three decimal places.

Explanation:

Step1: Calculate non - library students

There are $55 - 13=42$ students who do not volunteer at the library.

Step2: Calculate first - selection probability

The probability that the first - selected student does not volunteer at the library is $\frac{42}{55}$.

Step3: Calculate second - selection probability

Since one student has been selected without replacement, there are 54 students left and 41 non - library students left. So the probability that the second - selected student does not volunteer at the library is $\frac{41}{54}$.

Step4: Calculate third - selection probability

After two students have been selected, there are 53 students left and 40 non - library students left. So the probability that the third - selected student does not volunteer at the library is $\frac{40}{53}$.

Step5: Calculate the overall probability

The probability that none of the three students volunteer at the library is the product of the above probabilities: $P=\frac{42}{55}\times\frac{41}{54}\times\frac{40}{53}=\frac{42\times41\times40}{55\times54\times53}=\frac{68880}{157410}\approx0.438$

Answer:

$0.438$