Question

determine the probability that at least 2 people in a room of 14 people share the same birthday, ignoring leap years and assuming each birthday is equally likely, by answering the following questions.

(a) compute the probability that 14 people have different birthdays.
(b) use the result of part a to compute the probability that at least 2 people out of 14 share the same birthday.

(a) the probability that 14 people have different birthdays is 0.777.
(round to three decimal places as needed.)

(b) the probability that at least 2 people out of the 14 share a birthday is .
(round to three decimal places as needed.)

Explanation:

Calculate probability of different birthdays

Using the Birthday Problem Probability knowledge point

$$ P(\text{all different}) = \frac{365}{365} \times \frac{364}{365} \times \dots \times \frac{365 - 14 + 1}{365} \approx 0.777 $$

Apply the complement rule

Using the Birthday Problem Probability knowledge point

$$ P(\text{at least 2 share}) = 1 - P(\text{all different}) = 1 - 0.777 = 0.223 $$

Answer:

(b) The probability that at least 2 people out of the 14 share a birthday is <blank>0.223</blank>