Final answer:
The probability of at least one birthday match can be found by subtracting the probability of no birthday matches from 1. The probability of no matches decreases with each additional person added. The closest given answer to the calculated probability for a typical group size is 0.500.
Step-by-step explanation:
The question asks to find the probability of at least one birthday match among a group of people. This problem is a classic application of the complementary probability technique and involves the concept of probability in combinatorics.
To solve this, we calculate the probability of there being no birthday matches and subtract it from 1. The probability that the first person does not share a birthday with anyone else is 1 (365/365), the second person has a (364/365) chance of not sharing a birthday with the first person, the third person has a (363/365) chance of not sharing with the first two people, and so on. If we denote the number of people as n, the probability of no match with n people is:
∑(from i=0 to n-1) ((365-i)/365)
Using this approach for a typical group of 23 people, the probability of no match is approximately 0.493, and therefore the probability of at least one match is 1 - 0.493 = 0.507. The closest answer choice is (C) 0.500, which is roughly half and the intuitive answer considering there are 365 possible birthdays and a group size that introduces a significant chance of a match.