Let's solve this step-by-step.
### Step 1: Determine the Total Number of Balls
There are 2 sets of balls, and each set has 16 balls. Therefore, the total number of balls is:
[tex]\[ 2 \times 16 = 32 \][/tex]
### Step 2: Calculate the Total Number of Ways to Choose 2 Balls
To find the total number of ways to choose 2 balls from 32, we use the binomial coefficient, also known as "n choose k":
[tex]\[ \binom{32}{2} = \frac{32!}{2!(32-2)!} \][/tex]
This simplifies to:
[tex]\[ \binom{32}{2} = \frac{32 \times 31}{2 \times 1} = 496 \][/tex]
### Step 3: Determine the Number of Favorable Outcomes
We are looking for the number of ways to choose 2 balls that have the same number. There are 16 different numbers, and each number has 2 corresponding balls (one in each set). For each number, there is only 1 way to choose both balls with the same number:
[tex]\[ \text{Number of favorable outcomes} = 16 \][/tex]
### Step 4: Calculate the Probability
Finally, we find the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
[tex]\[ P(\text{same number}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \][/tex]
[tex]\[ P(\text{same number}) = \frac{16}{496} \][/tex]
This can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 16:
[tex]\[ P(\text{same number}) = \frac{1}{31} \][/tex]
In decimal form, this fraction is approximately:
[tex]\[ P(\text{same number}) \approx 0.032258 \][/tex]
### Conclusion
The probability that the two balls have the same number is [tex]\(\frac{1}{31}\)[/tex], or approximately 0.032258 when rounded to the nearest millionth.
