Question

c if four people are selected at random, find the probability that at least two of them have the same birthday. (round to three decimal places as needed.)

Explanation:

Step1: Calculate the probability of all - different birthdays

The first person can have a birthday on any of the 365 days of the year. 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 the fourth person on one of the remaining 362 days.
The probability that all four people have different birthdays is $P(\text{all different})=\frac{365}{365}\times\frac{364}{365}\times\frac{363}{365}\times\frac{362}{365}=\frac{365\times364\times363\times362}{365^{4}}\approx0.986$.

Step2: Calculate the probability of at - least two having the same birthday

The probability that at least two people have the same birthday is the complement of the event that all people have different birthdays.
Let $P(X\geq2)$ be the probability that at least two people have the same birthday. Then $P(X\geq2)=1 - P(\text{all different})$.
$P(X\geq2)=1-\frac{365\times364\times363\times362}{365^{4}}\approx1 - 0.986 = 0.014$.

Answer:

$0.014$