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Let's break this problem down step by step:

### Total Balls
We know the total number of balls is 210, and they are divided equally among three colors: red, blue, and green.

- Number of balls of each color:
  \[
  \frac{210}{3} = 70
  \]
  So, there are 70 red balls, 70 blue balls, and 70 green balls.

### Boxes and Their Capacities
There are two types of boxes:
- **Size A**: 6 boxes, each fitting 20 balls.
  \[
  6 \times 20 = 120 \text{ balls in size A boxes}
  \]
- **Size B**: 6 boxes, each fitting 15 balls.
  \[
  6 \times 15 = 90 \text{ balls in size B boxes}
  \]

### Probability of Selecting a Ball in a Size B Box
There are 90 balls in size B boxes out of a total of 210 balls. So, the probability that a randomly chosen ball is in a size B box is:
\[
P(\text{Size B}) = \frac{90}{210} = \frac{3}{7}
\]

### Probability of a Red Ball
Since the balls are equally distributed among the colors, the probability that a randomly chosen ball is red is:
\[
P(\text{Red}) = \frac{70}{210} = \frac{1}{3}
\]

### Combined Probability
To find the probability that a randomly chosen ball is **both red** and **in a size B box**, we multiply the two independent probabilities:
\[
P(\text{Red and Size B}) = P(\text{Red}) \times P(\text{Size B}) = \frac{1}{3} \times \frac{3}{7} = \frac{1}{7}
\]

Thus, the probability that a given ball is both red and shipped in a size B box is \( \frac{1}{7} \).

Would you like further clarification or details? Here are some related questions:

1. How would the probability change if the boxes had different proportions of balls?
2. What if the number of balls in size A and B boxes were different?
3. Can we calculate the expected number of red balls in size B boxes?
4. What if there were more colors, would the method change?
5. How would this problem differ if the balls were not equally distributed by color?

**Tip**: Always break down the problem into smaller steps, especially when dealing with probability, to avoid missing details.

A company produces equal numbers of red, blue, and green balls. A total of 210 balls are shipped in 6 size A boxes, which each fit 20 balls, and in 6 size B boxes, which each fit 15 balls. If the balls are packed randomly to fill all the boxes, what is the probability that a given ball is both red and shipped in a size B box?

This problem involves calculating the probability that a randomly chosen ball is both red and shipped in a size B box, given an equal distribution of red, blue, and green balls across different-sized boxes. The solution explains key probability concepts and uses the multiplication rule for independent events. Suitable for Grades 9-11.

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