**A bag contains 6 red marbles, 6 white marbles, and 8 blue marbles. You draw 4 marbles out at random, without replacement.**
*Round your answers to 4 decimal places as needed.*

**What is the probability that all the marbles are red?**
0.0031Correct

**What is the probability that exactly two of the marbles are red?**
0.2812Incorrect

**What is the probability that none of the marbles are red?**
0.2066Correct

Submit QuestionQuestion 8

Let's break down the problem and figure out the correct answer for the probability that exactly two of the marbles are red.

### Given:
- Total marbles: \(6\) red, \(6\) white, \(8\) blue → Total = \(6 + 6 + 8 = 20\) marbles.
- You draw 4 marbles at random, without replacement.

### To find: Probability that exactly 2 marbles are red.

We will use **combinations** since the order in which the marbles are drawn does not matter.

#### Step 1: Total possible outcomes
The total number of ways to choose 4 marbles from 20 is:
\[
\text{Total combinations} = \binom{20}{4} = \frac{20!}{4!(20-4)!} = 4845
\]

#### Step 2: Favorable outcomes for exactly 2 red marbles
We want to pick exactly 2 red marbles, and the remaining 2 marbles should be non-red (either white or blue). We calculate these combinations separately:

- **Ways to choose 2 red marbles out of 6 red marbles**:
\[
\binom{6}{2} = \frac{6!}{2!(6-2)!} = 15
\]

- **Ways to choose 2 non-red marbles out of the 14 non-red marbles (6 white + 8 blue)**:
\[
\binom{14}{2} = \frac{14!}{2!(14-2)!} = 91
\]

#### Step 3: Total favorable outcomes
The total number of favorable outcomes for exactly 2 red marbles is:
\[
\text{Favorable outcomes} = \binom{6}{2} \times \binom{14}{2} = 15 \times 91 = 1365
\]

#### Step 4: Probability calculation
The probability of drawing exactly 2 red marbles is the ratio of favorable outcomes to the total possible outcomes:
\[
P(\text{exactly 2 red}) = \frac{1365}{4845} \approx 0.2816
\]

### Conclusion:
The correct probability that exactly two of the marbles are red is **0.2816**, not **0.2812**.

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Do you want more details or have any questions?

Here are five related questions you could explore:
1. What is the probability of drawing at least one red marble?
2. What is the probability that exactly one marble is red?
3. What is the probability of drawing all white marbles?
4. What is the probability of drawing two blue marbles and two white marbles?
5. How does the probability change if marbles are drawn with replacement?

**Tip**: Always double-check your combination calculations to ensure accuracy!

A bag contains 6 red marbles, 6 white marbles, and 8 blue marbles. You draw 4 marbles out at random, without replacement. What is the probability that all the marbles are red? What is the probability that exactly two of the marbles are red? What is the probability that none of the marbles are red?

Explore how to calculate the probability of drawing marbles from a bag containing 6 red, 6 white, and 8 blue marbles. Learn step-by-step how to solve for the probability of drawing all red marbles, exactly two red marbles, and no red marbles, using combinations and probability formulas. Suitable for students in Grades 10-12.

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