Step 1 :The problem is asking for the probability that at least two people out of 19 selected at random have the same birthday. This is equivalent to 1 minus the probability that all 19 people have different birthdays.
Step 2 :To calculate the probability that all 19 people have different birthdays, we can use the same logic as in part a. The first person can have any birthday, so the probability is \( \frac{365}{365} \). The second person must have a different birthday, so the probability is \( \frac{364}{365} \). This continues until the 19th person, who has 347 possible birthdays left, so the probability is \( \frac{347}{365} \).
Step 3 :The probability that all 19 people have different birthdays is then the product of these probabilities.
Step 4 :Finally, the probability that at least two people have the same birthday is 1 minus this product.
Step 5 :Let's calculate this. The probability that all 19 people have different birthdays is 0.620881473968463. The probability that at least two people have the same birthday is 0.37911852603153695.
Step 6 :Final Answer: The probability that at least two out of 19 people selected at random have the same birthday is approximately \(\boxed{0.379}\).