Final answer:
The probability that at least two of the nine justices of the U.S. Supreme Court have the same birthday can be calculated using the concept of complementary probability.
Step-by-step explanation:
The probability that at least two of the nine justices of the U.S. Supreme Court have the same birthday can be calculated using the concept of complementary probability. We can find the probability of all justices having different birthdays and subtract it from 1 to get the probability of at least two having the same birthday.
The probability of the first justice having a unique birthday is 1 (as there are no other birthdays to match with). The probability of the second justice having a unique birthday is 364/365 (assuming a non-leap year), as there is one birthday to avoid. Similarly, the probability for the third justice having a unique birthday is 363/365, and so on.
Using these probabilities, we can calculate the probability of all justices having different birthdays:
P(all different) = (1) * (364/365) * (363/365) * ... * (357/365)
Finally, we can subtract this probability from 1 to get the probability of at least two justices having the same birthday:
P(at least two same) = 1 - P(all different)