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: Find total non - library students

First, we need to find the number of students not volunteering at the library. The total number of students is 55, and the number of students at the library is 13. So the number of non - library students is \(55 - 13=42\).

Step2: Calculate the probability for the first selection

When we select the first student, the probability that the student is not from the library is the number of non - library students divided by the total number of students. So \(P_1=\frac{42}{55}\).

Step3: Calculate the probability for the second selection

After selecting one non - library student (without replacement), the number of non - library students left is \(42 - 1 = 41\) and the total number of students left is \(55 - 1=54\). So the probability that the second student is also not from the library is \(P_2=\frac{41}{54}\).

Step4: Calculate the probability for the third selection

After selecting two non - library students (without replacement), the number of non - library students left is \(41- 1=40\) and the total number of students left is \(54 - 1 = 53\). So the probability that the third student is also not from the library is \(P_3=\frac{40}{53}\).

Step5: Calculate the combined probability

Since the selections are independent events (without replacement, we use the multiplication rule for dependent events), the probability that all three students are not from the library is \(P = P_1\times P_2\times P_3=\frac{42}{55}\times\frac{41}{54}\times\frac{40}{53}\).

First, calculate \(\frac{42}{55}\times\frac{41}{54}=\frac{42\times41}{55\times54}=\frac{1722}{2970}=\frac{287}{495}\) (simplifying the fraction by dividing numerator and denominator by 6). Then \(\frac{287}{495}\times\frac{40}{53}=\frac{287\times40}{495\times53}=\frac{11480}{26235}\approx0.437\)

Answer:

\(0.437\)