There are 2
2
sets of balls numbered 1
1
through 17
17
placed in a bowl. If 2
2
balls are randomly chosen without replacement, find the probability that the balls have the same number. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

We are given that there are two sets of balls numbered 1 through 17 placed in a bowl, making a total of \( 2 \times 17 = 34 \) balls. We are asked to find the probability of choosing two balls that have the same number when drawing without replacement.

### Step 1: Total number of ways to choose 2 balls
The total number of ways to choose 2 balls from 34 is given by the combination formula:

\[
\binom{34}{2} = \frac{34 \times 33}{2} = 561
\]

### Step 2: Number of favorable outcomes
To have the two balls with the same number, the two balls chosen must be one from the first set and one from the second set (since each number has two balls). There are 17 different numbers, and for each number, there is exactly 1 pair of matching balls (one from each set).

Thus, the number of favorable outcomes is 17.

### Step 3: Probability calculation
The probability is the ratio of favorable outcomes to the total number of outcomes:

\[
P = \frac{17}{561}
\]

This fraction is already in its lowest terms. To express it as a decimal, divide 17 by 561:

\[
P \approx 0.030303
\]

### Final Answer:
The probability that the two randomly chosen balls have the same number is:

\[
P \approx 0.030303
\]

or \( \frac{17}{561} \).

Would you like further details or have any questions?

Here are 5 related questions to explore this topic further:
1. How would the probability change if there were 3 sets of balls instead of 2?
2. What if the total number of balls were increased to 50? How would that affect the outcome?
3. Can you generalize this probability formula for any number of sets and any number of balls?
4. What is the probability of selecting two balls with consecutive numbers instead?
5. How does the probability change if we return the first ball before selecting the second?

**Tip:** When calculating probabilities, ensure that the total number of possible outcomes reflects the correct scenario, such as "without replacement" or "with replacement."

There are 2 sets of balls numbered 1 through 17 placed in a bowl. If 2 balls are randomly chosen without replacement, find the probability that the balls have the same number. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

Find the probability of selecting two balls with the same number from a bowl containing two sets of balls numbered 1 through 17. This problem covers key concepts in probability and combinatorics, including the use of combinations and the probability formula. Suitable for Grades 9-12.

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