We are given that the probability of a baby being a boy is $P(\text{boy}) = 0.513$, which means the probability of a baby being a girl is:

$P(\text{girl}) = 1 - P(\text{boy}) = 1 - 0.513 = 0.487$

Now, the problem asks for the probability of having at least one girl among the next nine randomly selected births. The complementary event to this is having no girls, which means all nine babies are boys. We can first calculate the probability of all nine babies being boys, and then subtract that from 1 to find the probability of having at least one girl.

The probability of all nine babies being boys is:

$P(\text{all boys}) = (0.513)^9$

Then, the probability of at least one girl is:

$P(\text{at least one girl}) = 1 - P(\text{all boys})$

Let me calculate that for you.The probability that at least one of the nine randomly selected births is a girl is $0.998$, rounded to three decimal places.

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:

1. How do you calculate the probability of multiple independent events?
2. What is the significance of complementary probability in probability theory?
3. How does the binomial distribution apply to problems like this?
4. What is the probability of having exactly 2 girls in the next 9 births?
5. Can we model this situation using a normal approximation for large samples?

Tip: Always think of using the complement rule when solving "at least" type probability problems to simplify the calculation.