Question

Our class has 37 students.

i. Without any calculations, what would you guess is the chance that two people in our class share the

same birthday? In other words, do you think it is likely that two people share a birthday?

ii. How many possible pairs of students are there?

iii. What is the probability that any random pair of students has the same birthday?

iv. What is the probability that all of the pairs have different birthdays?

v. So, what is the probability that there is at least one shared birthday in our class?

Answer 1

Solution:

i) Since, there are too many days in a year that's why I think it is not very likely that two people share a birthday.

ii) We have total 37 students and 2 students will make a pair.

Total number of way of selecting k people from n people is given by,

Hence, total number of possible pairs of students are,

Total number of possible pairs of students are 666.

iii) We shall assume that there is no leap year and hence, 365 days are there in a year.

Let a random pair of students named as "A" and "B". We shall consider A as first student and B as second student.

First student A has total 365 days for his birthday out of 365 days of a year.

Probability that student A has birthday on any day of the year will be 365/365 = 1.

Now, student B should have the same birthday as the birthday of student A. Hence, he has only 1 day for his birthday out of 365 days of a year.

Probability that student B will have the same birthday as student A = 1/365

Hence, probability that student A and B have the same birthday is 1 × (1/365) = 1/365 = 0.00274.

Hence, probability that any random pair of students has the same birthday is 0.00274.

iv) First of all we shall obtain the probability that a random pair of student has the different birthday.

Since, the total probability is equal to 1, therefore,

probability that a random pair of student has the different birthday = 1 - probability that a random pair of student has the same birthday

In part (iii) we have obtained the probability that a random pair of student has the same birthday is 1/365.

Hence, probability that a random pair of student has the different birthday = 1 - (1/365) = 364/365

Also we have obtained in part (ii) that total number of possible pairs is 666.

Hence, probability that all of the pairs have different birthdays is,

Probability that all of the pairs have different birthdays is 0.16087.

v) Probability that at least one shared birthday in our class = 1 - probability that no pair have the same birthday

i.e.  Probability that at least one shared birthday in our class = 1 - probability that all pairs have the different birthdays.

In part (iv) we have obtained that probability that all pairs have different birthdays = 0.16087

Hence, Probability that at least one shared birthday in our class = 1 - 0.16087 = 0.83913

Probability that at least one shared birthday in our class is 0.83913.