Question

an ice - chest contains 8 cans of apple juice, 4 cans of grape juice, 4 cans of orange juice, and 5 cans of mango juice. suppose that you reach into the cooler and randomly select three cans in succession. find the probability of selecting three cans of grape juice. the probability of selecting from the ice - chest three cans of grape juice is (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate total number of cans

The total number of cans is \(8 + 4+4 + 5=21\) cans.

Step2: Use combination formula for probability

The number of ways to choose 3 cans out of 21 is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 21\) and \(r=3\). So \(C(21,3)=\frac{21!}{3!(21 - 3)!}=\frac{21\times20\times19}{3\times2\times1}=1330\).
The number of ways to choose 3 cans of grape - juice. There are 4 cans of grape - juice, and the number of ways to choose 3 cans out of 4 is \(C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=4\).

Step3: Calculate the probability

The probability \(P\) of selecting 3 cans of grape - juice is the number of favorable outcomes divided by the number of total outcomes. So \(P=\frac{C(4,3)}{C(21,3)}=\frac{4}{1330}=\frac{2}{665}\).

Answer:

\(\frac{2}{665}\)