Question

an ice chest contains 8 cans of apple juice, 6 cans of grape juice, 5 cans of orange juice, and 8 cans of mango juice. suppose that you reach into the container and randomly select three cans in succession. find the probability of selecting three cans of mango juice. the probability of selecting from the ice chest three cans of mango juice is (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate total number of cans

Total number of cans \(n = 8 + 6+5 + 8=27\)

Step2: Use combination formula for probability

The formula for probability of selecting \(r\) items from \(n\) items and \(k\) items from \(m\) items is \(P=\frac{C(m,r)}{C(n,r)}\)
Here, \(m = 8\) (mango juice cans), \(r = 3\), \(n=27\)
Combination formula \(C(a,b)=\frac{a!}{b!(a - b)!}\)
\(C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8\times7\times6}{3\times2\times1}=56\)
\(C(27,3)=\frac{27!}{3!(27 - 3)!}=\frac{27\times26\times25}{3\times2\times1}=2925\)

Step3: Calculate the probability

\(P=\frac{56}{2925}=\frac{8}{417.857}\approx\frac{8}{418}=\frac{4}{209}\)

Answer:

\(\frac{4}{209}\)