An urn contains 8 red marbles, 10 white marbles, and 9 blue marbles marbles. A child randomly selects three (without replacement) from the urn. Round to four decimal places.
Find the probability all three marbles are the same color. 


To solve the problem, we need to calculate the probability that all three marbles selected are of the same color (either all red, all white, or all blue). We will use the concept of combinations to calculate this.

### Step 1: Total number of marbles
The total number of marbles is:
\[
8 \, (\text{red}) + 10 \, (\text{white}) + 9 \, (\text{blue}) = 27 \, (\text{total marbles})
\]

### Step 2: Total number of ways to choose 3 marbles from 27
The total number of ways to choose 3 marbles from 27 marbles is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of marbles, and \( r \) is the number of marbles chosen (in this case, 3):
\[
\text{Total ways} = \binom{27}{3} = \frac{27 \times 26 \times 25}{3 \times 2 \times 1} = 2925
\]

### Step 3: Number of favorable outcomes (all same color)
Now, we calculate the number of ways to select 3 marbles of the same color for each color:

1. **All red**: There are 8 red marbles, and the number of ways to choose 3 red marbles from 8 is:
   \[
   \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
   \]

2. **All white**: There are 10 white marbles, and the number of ways to choose 3 white marbles from 10 is:
   \[
   \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
   \]

3. **All blue**: There are 9 blue marbles, and the number of ways to choose 3 blue marbles from 9 is:
   \[
   \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84
   \]

### Step 4: Total favorable outcomes
The total number of favorable outcomes (all three marbles are the same color) is the sum of the combinations for red, white, and blue:
\[
56 + 120 + 84 = 260
\]

### Step 5: Probability
The probability of selecting three marbles of the same color is the ratio of favorable outcomes to the total number of possible outcomes:
\[
\text{Probability} = \frac{260}{2925} \approx 0.0889
\]

### Final Answer:
The probability that all three marbles are the same color is approximately \( \boxed{0.0889} \).

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the probability of selecting exactly two red marbles and one white marble?
2. How does the probability change if the selection is done with replacement?
3. What is the probability of selecting one marble of each color?
4. If 4 marbles were selected instead of 3, what is the probability of getting all marbles of the same color?
5. How does the total probability change if one additional red marble is added to the urn?

**Tip:** When calculating probabilities involving combinations, always be careful with the "without replacement" or "with replacement" condition as they lead to different counting methods.

An urn contains 8 red marbles, 10 white marbles, and 9 blue marbles. A child randomly selects three (without replacement) from the urn. Find the probability all three marbles are the same color.

This problem demonstrates how to calculate the probability of selecting three marbles of the same color from an urn containing 8 red, 10 white, and 9 blue marbles. The solution involves combinatorics and probability formulas, suitable for students in Grades 10-12.

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