3. A grocery store is reviewing its restocking policies and has analyzed the number of half-gallon containers of orange juice sold each day for the past month. The data are given below.
Number sold morning afternoon evening
0-19 3 8 2
20-39 3 4 3
40-59 12 8 4
60-79 4 9 9
80-99 5 3 6
100 or more 3 0 6

a. What is the probability that on a randomly selected day the number of cartons of orange juice sold in the evening is between 80 and 99?
b. What is the probability that 39 or fewer cartons were sold during a randomly selected afternoon?
c. What is the probability that either 0-19 or 100 or more cartons were sold in a randomly selected morning?

Let's break down each part of the problem using probability principles and the given data. The goal is to determine the probability for each scenario, which is computed by dividing the favorable outcomes by the total number of outcomes.

### Total Data
We can first sum up the total number of days in each time period (morning, afternoon, and evening) to help us with the probabilities. Here's the full data layout:

| Number Sold | Morning | Afternoon | Evening |
|-------------|---------|-----------|---------|
| 0-19        | 3       | 8         | 2       |
| 20-39       | 3       | 4         | 3       |
| 40-59       | 12      | 8         | 4       |
| 60-79       | 4       | 9         | 9       |
| 80-99       | 5       | 3         | 6       |
| 100+        | 3       | 0         | 6       |

Now we sum the total number of days in each time period:

- **Morning**: \( 3 + 3 + 12 + 4 + 5 + 3 = 30 \)
- **Afternoon**: \( 8 + 4 + 8 + 9 + 3 + 0 = 32 \)
- **Evening**: \( 2 + 3 + 4 + 9 + 6 + 6 = 30 \)

#### a. What is the probability that on a randomly selected day the number of cartons of orange juice sold in the evening is between 80 and 99?

In the evening, the number of days where 80-99 cartons were sold is **6**.

- Total number of days in the evening: 30
- Probability = \( \frac{6}{30} = 0.2 \)

Thus, the probability is \( \boxed{0.2} \) or 20%.

#### b. What is the probability that 39 or fewer cartons were sold during a randomly selected afternoon?

To find this, we need to consider the number of days where either 0-19 or 20-39 cartons were sold in the afternoon:

- Days with 0-19 cartons: 8
- Days with 20-39 cartons: 4
- Total favorable outcomes = \( 8 + 4 = 12 \)

- Total number of days in the afternoon: 32
- Probability = \( \frac{12}{32} = 0.375 \)

Thus, the probability is \( \boxed{0.375} \) or 37.5%.

#### c. What is the probability that either 0-19 or 100 or more cartons were sold in a randomly selected morning?

For the morning, we need to find the number of days where either 0-19 or 100+ cartons were sold:

- Days with 0-19 cartons: 3
- Days with 100+ cartons: 3
- Total favorable outcomes = \( 3 + 3 = 6 \)

- Total number of days in the morning: 30
- Probability = \( \frac{6}{30} = 0.2 \)

Thus, the probability is \( \boxed{0.2} \) or 20%.

### Summary of Answers:
- **a.** The probability that 80-99 cartons were sold in the evening is \( 0.2 \) or 20%.
- **b.** The probability that 39 or fewer cartons were sold in the afternoon is \( 0.375 \) or 37.5%.
- **c.** The probability that either 0-19 or 100+ cartons were sold in the morning is \( 0.2 \) or 20%.

Would you like more details or explanations on any part?

### Related Questions:
1. What is the probability that exactly 60-79 cartons were sold in the evening?
2. What is the probability that more than 59 cartons were sold during a randomly selected afternoon?
3. What is the combined probability that between 40-79 cartons were sold in a morning or afternoon?
4. How many days in total were cartons sold in the range of 40-59 across all time periods?
5. What is the overall probability that 0-39 cartons were sold across all three time periods?

### Tip:
When calculating probabilities, always double-check the total outcomes and ensure that your favorable outcomes align with the question's conditions.

A grocery store is reviewing its restocking policies and has analyzed the number of half-gallon containers of orange juice sold each day for the past month. The data are given below. Number sold morning afternoon evening 0-19 3 8 2 20-39 3 4 3 40-59 12 8 4 60-79 4 9 9 80-99 5 3 6 100 or more 3 0 6. a. What is the probability that on a randomly selected day the number of cartons of orange juice sold in the evening is between 80 and 99? b. What is the probability that 39 or fewer cartons were sold during a randomly selected afternoon? c. What is the probability that either 0-19 or 100 or more cartons were sold in a randomly selected morning?

Explore probability analysis using real-world data from a grocery store's orange juice sales. Calculate the likelihood of selling between 80-99 cartons in the evening, fewer than 39 cartons in the afternoon, or specific ranges in the morning. This problem covers core probability and statistics concepts, ideal for Grades 9-12.

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