Question

an ice chest contains 7 cans of apple juice, 8 cans of grape juice, 4 cans of orange juice, and 6 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 apple juice. the probability of selecting from the ice chest three cans of apple juice is (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate total number of cans

The total number of cans is \(7 + 8+4 + 6=25\) cans.

Step2: Use combination formula for probability

The number of ways to choose 3 cans out of 25 is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 25\) and \(r=3\), so \(C(25,3)=\frac{25!}{3!(25 - 3)!}=\frac{25\times24\times23}{3\times2\times1}=2300\).
The number of ways to choose 3 cans of apple - juice out of 7 is \(C(7,3)=\frac{7!}{3!(7 - 3)!}=\frac{7\times6\times5}{3\times2\times1}=35\).

Step3: Calculate probability

The probability \(P\) of selecting 3 cans of apple - juice is the number of favorable outcomes divided by the number of total outcomes. So \(P=\frac{C(7,3)}{C(25,3)}=\frac{35}{2300}=\frac{7}{460}\).

Answer:

\(\frac{7}{460}\)