Question

show that if 23 people are selected at random, the probability that at least 2 of them have the same birthday is greater than 1/2. the probability that none of the 23 people share a birthday is . the probability that at least 2 of 23 people have the same birthday is . the probability that there exists people with the same birthday is (round to three decimal places as needed.)

Explanation:

Step1: Calculate probability of no - shared birthdays

The first person can have a birthday on any of the 365 days. The second person must have a birthday on one of the remaining 364 days, the third person on one of the remaining 363 days, and so on. For 23 people, the probability $P(\text{no shared birthdays})$ is given by the formula:

$$P(\text{no shared birthdays})=\frac{365}{365}\times\frac{364}{365}\times\frac{363}{365}\times\cdots\times\frac{365 - 22}{365}$$

$$P(\text{no shared birthdays})=\prod_{k = 0}^{22}\frac{365 - k}{365}$$

$$P(\text{no shared birthdays})\approx0.493$$

Step2: Calculate probability of at least 2 people sharing birthdays

The probability that at least 2 people share a birthday is the complement of the event that none of them share a birthday. Let $P(\text{at least 2 share})$ be the probability that at least 2 people share a birthday. Then $P(\text{at least 2 share})=1 - P(\text{no shared birthdays})$.

$$P(\text{at least 2 share})=1-0.493 = 0.507$$

Answer:

The probability that none of the 23 people share a birthday is approximately $0.493$. The probability that at least 2 of 23 people have the same birthday is approximately $0.507$.