This exercise is on probabilties and coincidence of shared birthdays. Complete parts (a) through (e) below
a. If two people are selected at random, the probability that they do not have the same birthday (day and month) is $\frac{365}{365} \cdot \frac{364}{365}$. Explain why this is so. (Ignore leap years and assume 365 days in a year)
The first person can have any birthday, so they can have a birthday on 365 of the 365 days. In order for the second person to not have the same birthday they must have one of the 364 remaining birthdays (Type whole numbers)
b. If five people are selected at randorn, find the probability that they all have different birthdays
The probability that they all have different birthdays is 0.973
(Round to three decimal places as needed)
c. If five people are selected at random, find the probability that at least two of them have the same biihday
The probability that at least two of them have the same birthday is 0.027
(Round to three decimal places as needed)
d. If 19 people are selected at random, find the probability that at least 2 of them have the same birthday
The probability that at least two of them have the same birthday is $\square$
(Round to three decimal places as needed)
Answer
Final Answer: The probability that at least two out of 19 people selected at random have the same birthday is approximately \(\boxed{0.379}\).
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 \(\boxed{0.379}\).
