An ice chest contains six cans of apple juice, four cans of grape juice, seven cans of
orange juice, and eight 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.
The probability of selecting no apple juice from the ice chest is
(Type an integer or a simplified fraction.)

Question

UpStudy AI · Accepted Answer

1. Total number of cans is  
   $$
   6 + 4 + 7 + 8 = 25.
   $$

2. Number of non-apple juice cans is  
   $$
   4 + 7 + 8 = 19.
   $$

3. The probability that the first can selected is not apple juice is  
   $$
   \frac{19}{25}.
   $$

4. After one non-apple can is removed, there are 18 non-apple cans left out of 24 cans. So, the probability for the second can is  
   $$
   \frac{18}{24}.
   $$

5. After removing a second non-apple can, there are 17 non-apple cans left out of 23 cans. Thus, the probability for the third can is  
   $$
   \frac{17}{23}.
   $$

6. The overall probability of selecting three non-apple juice cans is the product of these probabilities:  
   $$
   \frac{19}{25} \times \frac{18}{24} \times \frac{17}{23} = \frac{19 \times 18 \times 17}{25 \times 24 \times 23}.
   $$

7. Calculate the products in the numerator and denominator:  
   $$
   \text{Numerator} = 19 \times 18 \times 17 = 5814,
   $$  
   $$
   \text{Denominator} = 25 \times 24 \times 23 = 13800.
   $$

8. Simplify the fraction:  
   Divide numerator and denominator by 2,  
   $$
   \frac{5814}{13800} = \frac{2907}{6900}.
   $$  
   Divide numerator and denominator by 3,  
   $$
   \frac{2907}{6900} = \frac{969}{2300}.
   $$

The probability of selecting no apple juice is  
$$
\boxed{\frac{969}{2300}}.
$$
The probability of selecting no apple juice is $$\frac{969}{2300}$$.

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