Question

A biology class has 28 students. Find the probability that at least two students in the class have the same birthday. For​ simplicity, assume that there are always 365 days in a year and that birth rates are constant throughout the year.​ (Hint: First, determine the probability that no 2 students have the same birthday and then apply the complementation​ rule.)

Answer 1

A biology class has 28 students. Now we want to find the probability that at least two students in the class have the same birthday.

A: Event denoting that at least two students in the class have the same birthday.

So our required probability P(A) is equivalently given by,  

=> P(at least two students in the class have the same birthday) = 1 - (Probability of no students share same birthday)

Now we find the probability i.e. probability that no students shares the same birthday out of a class of 28 students.

The first student can be born on any day of the year, so the first students probability is 365/365.

The second student is now limited to 364 possible days, so the second students probability is 364/365.

The third student is now limited to 363 possible days, so the third students probability is  363/365.

This pattern continues to our last student i.e. 28th student has a probability of 338/365

Now we multiply all 28 probabilities together and we get,

So we get,

So our required probability is given by,

  [ Round to four decimal places ]

Answer:- Probability that at least two students in the class have the same birthday is 0.6545