To solve this problem, we determine the total number of possible outcomes and favorable outcomes where exactly two of the balls are the same color. Here is the step-by-step process:

---

Step 1: Total Number of Ways to Choose 3 Balls

The bag contains $8$ blue balls and $2$ red balls, so there are $10$ balls in total. The total number of ways to choose $3$ balls from $10$ is:

$\binom{10}{3} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120$

---

Step 2: Conditions for Exactly Two Balls of the Same Color

For exactly two balls to be of the same color, there are two possible scenarios:

1. Two blue balls and one red ball
2. Two red balls and one blue ball

---

Scenario 1: Two Blue Balls and One Red Ball

The number of ways to choose $2$ blue balls out of $8$ is:

$\binom{8}{2} = \frac{8 \cdot 7}{2} = 28$

The number of ways to choose $1$ red ball out of $2$ is:

$\binom{2}{1} = 2$

Thus, the total number of favorable outcomes for this scenario is:

$28 \cdot 2 = 56$

---

Scenario 2: Two Red Balls and One Blue Ball

The number of ways to choose $2$ red balls out of $2$ is:

$\binom{2}{2} = 1$

The number of ways to choose $1$ blue ball out of $8$ is:

$\binom{8}{1} = 8$

Thus, the total number of favorable outcomes for this scenario is:

$1 \cdot 8 = 8$

---

Step 3: Total Favorable Outcomes

The total number of favorable outcomes is the sum of the outcomes from both scenarios:

$56 + 8 = 64$

---

Step 4: Probability

The probability is the ratio of favorable outcomes to total outcomes:

$P(\text{exactly two balls of the same color}) = \frac{64}{120} = \frac{8}{15}$

---

Final Answer:

The probability that exactly two of the balls are the same color is:

$\boxed{\frac{8}{15}}$

---

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

1. What is the probability of selecting three blue balls?
2. What is the probability of selecting three red balls?
3. How does the probability change if there are more red balls in the bag?
4. Can you explain combinations ($\binom{n}{r}$) in detail?
5. How would the result change if four balls were selected instead of three?

Tip: When calculating probabilities, ensure you correctly identify and account for all possible scenarios that meet the conditions.