Question

this exercise is on probabilities and coincidence of shared birthdays. complete parts (a) through (e) below (round to three decimal places as needed.) c. if four people are selected at random, find the probability that at least two of them have the same birthday the probability that at least two of them have the same birthday is 0.014 (round to three decimal places as needed.) d. if 14 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 0.212 (round to three decimal places as needed.) e. 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 than the probability of all 23 not sharing a birthday (round to three decimal places as needed.)

Explanation:

Step1: Calculate probability of no - shared birthdays

The probability that \(n\) people all have different birthdays. For \(n\) people, the first person can have a birthday on any of 365 days, the second person on \(365 - 1\) days, the third person on \(365-2\) days and so on. So the probability that \(n\) people have different birthdays is \(P(\text{no match})=\frac{365}{365}\times\frac{364}{365}\times\frac{363}{365}\times\cdots\times\frac{365-(n - 1)}{365}\).
For \(n = 23\), we have \(P(\text{no match})=\prod_{k = 0}^{22}\frac{365 - k}{365}=\frac{365!}{365^{23}(365 - 23)!}\)
\[

$$\begin{align*} P(\text{no match})&=\frac{365\times364\times\cdots\times343}{365^{23}}\\ &\approx0.493 \end{align*}$$

\]

Step2: Calculate probability of at - least one shared birthday

The probability that at least two people have the same birthday is the complement of the event that no two people have the same birthday. Let \(P(\text{match})\) be the probability that at least two people have the same birthday. Then \(P(\text{match})=1 - P(\text{no match})\).
Since \(P(\text{no match})\approx0.493\), then \(P(\text{match})=1 - 0.493 = 0.507\)

Answer:

The probability that none of the 23 people share a birthday is \(0.493\). The probability that at least 2 of 23 people have the same birthday is \(0.507\). The probability that there exist people with the same birthday is greater than the probability of all 23 not sharing a birthday.