A box contains 8 red balls and 2 blue balls. You draw 2 balls at random
(without replacement). What is the probability that the two balls have
different colors?

To solve this problem, let's first outline the necessary steps:

### Step 1: Identify the total number of possible outcomes.
Since you're drawing 2 balls from a total of 10 (8 red and 2 blue) without replacement, the total number of possible outcomes is the number of ways to choose 2 balls from 10. This can be calculated using the combination formula:

\[
\text{Total outcomes} = \binom{10}{2} = \frac{10 \times 9}{2} = 45
\]

### Step 2: Identify the favorable outcomes (drawing one red and one blue ball).
To have different colors, you must draw one red ball and one blue ball. The number of ways to do this is the product of choosing 1 red ball from 8 and 1 blue ball from 2:

\[
\text{Favorable outcomes} = \binom{8}{1} \times \binom{2}{1} = 8 \times 2 = 16
\]

### Step 3: Calculate the probability.
The probability of drawing two balls of different colors is the ratio of favorable outcomes to total outcomes:

\[
\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{16}{45}
\]

Thus, the probability that the two balls have different colors is:

\[
\boxed{\frac{16}{45}}
\]

Would you like further details on any of the steps, or do you have any questions?

### Related Questions:
1. What is the probability of drawing two red balls?
2. What is the probability of drawing two blue balls?
3. How would the probability change if the box had 3 blue balls?
4. What is the probability of drawing exactly one blue ball if you draw 3 balls?
5. How does the probability change if the balls are drawn with replacement?

**Tip:** When solving probability problems, identifying favorable outcomes using combinations is key to simplifying the calculations.

A box contains 8 red balls and 2 blue balls. You draw 2 balls at random (without replacement). What is the probability that the two balls have different colors?

Calculate the probability of drawing two balls of different colors from a box containing 8 red balls and 2 blue balls without replacement. This problem involves key concepts of probability and combinatorics, providing a step-by-step guide to solve it. Suitable for students in Grades 10-12.

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