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Chapter 11: Problem 48

An ice chest contains six cans of apple juice, eight cans of grape juice, four cans of orange juice, and two cans of mango juice. Suppose that you reach into the container and randomly select three cans in succession. Find the probability of selecting no apple juice.

Short Answer

Expert verified
The probability of not picking any apple juice is calculated by dividing the number of ways of picking 3 cans from the 14 non-apple juice cans (which can be denoted as \(C(14,3)\)) by the total number of ways of picking 3 cans out of 20 (denoted as \(C(20,3)\)). The answer will be found by evaluating these combinations and performing the division.

Step by step solution

01

Total Number of Ways to Pick 3 Cans

First of all, you would compute the total number of ways to choose three cans out of 20. This can be done by calculating the combination of picking 3 out of 20, which is denoted as \(C(20,3)\) and can be calculated by the formula \(C(n,r) = n! / r!(n-r)!\), where \(n\) is the total number of items, \(r\) is the number of items being chosen at a time, and \(!\) denotes factorial.
02

Calculate Number of Ways to Pick 3 Cans That Are Not Apple Juice

Next, you would calculate the number of ways to pick three cans where none of them is apple juice. Located in the chest there are 14 cans that are non-apple juice. Hence, the number of ways to pick 3 out of these 14 cans is \(C(14,3)\).
03

Calculate the Probability

Probability is defined as the number of desired outcomes divided by the total number of outcomes. Thus, to find the probability of drawing no apple juice, you would divide the combination of picking 3 out of 14 (non-apple juice cans) by the total combination of picking 3 out of 20. Mathematically this can be seen as \( P(\text{no apple juice}) = C(14,3) / C(20,3) \).

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