Question

Problem#5: The Birthday Problem (10pts) In your classroom there are 45 students, assume that none of them is born on February 29, and hence consider only common years (365 days) [do not consider leap years]. 1- Take a student randomly from the class, what is the probability that he have the same birthday as yours? 2- What is the probability that there is at least one student in the class having the same birthday as yours? 3-What is the probability that there is no repeated birthday date in the class (all the 45 birthdays being different)? Comment on the value you found 4- The instructor decides to nominate 5 students randomly to deliver a presentation next class, • How many groups (of five students) are possible? • What is the probability that you are among the selected students?

Answer 1

1) Probability that the student selected has the same birthday as me is computed here as:

= 1/365

= 0.0027

Therefore 0.0027 is the required probability here.

2) Probability that there is at least one student in the class with same birthday as me is computed here as:

= 1 - Probability that no student int he class has same birthday as me

= 1 - (364/365)⁴⁵

= 0.1161

Therefore 0.1161 is the required probability here.

3) Probability that all 45 students have got different birthdays

= Number of permutation of 365 days taken 45 at a time / Number of ways to assign birthdays to 45 students

Therefore 0.0590 is the required probability here.

4) Number of ways to select 5 students from the 365 students here is computed as:

= Combination of 45 people taken 5 at a time

These are the number of required ways here.

Probability that I am amongst the selected students is computed here as:

= Number of ways to select the remaining 4 students from 44 students / Total ways to select 5 students form 45 students

Therefore 0.1111 is the required probability here.