Question

Sourte and Harmide are friends. What is the probability that both have
1.same birthday
2.different birthday ?

Answer 1

Hint: Probability of an event A is P(A). It is equal to the ratio of favorable outcomes for A and total outcomes. We know that, for two independent events A and B, P(AB)= P(A) × P(B). If the probability of occurrence of an event A is x then the probability of occurrence of event Ac is 1-x.

Complete answer:
For simplicity we find the values of question 2 first and then for the 1st question.
Let A be the event corresponding to the birthday of Sourte and B be the event corresponding to the birthday of Harmide.
Let, the total number of days in a year is 365.
So, we have total outcomes related to A and B is 365.
Now, Sourte can have any birthday between 365 days.
So, the favorable outcomes corresponding to event A is 365.
So, P(A) \[ = \] P( birthday of Sourte)
\[ = \] \[\dfrac{{{\text{favorable outcomes for A}}}}{{{\text{total outcomes }}}} \]
\[ = \] \[\dfrac{{365}}{{365}}\]
Now, to have different birthdays, Harmide cannot have the same birthday which is the same as Sourte’s birthday.
So, the favorable outcomes corresponding to event B is 365\[ - \]1\[ = \]364.
So, P(B) \[ = \] \[ = \] P( birthday of Harmide) \[ = \] \[
\dfrac{{{\text{favorable outcomes for B}}}}{{{\text{total outcomes }}}} \]
\[ = \]\[\dfrac{{364}}{{365}}\]
Hence, probability of both of them having different birthdays \[ = \] P(AB)
  \[ = \] P(A) \[ \times \] P(B)
\[ = \]\[\dfrac{{365}}{{365}}\]\[ \times \]\[\dfrac{{364}}{{365}}\]
\[ = \] 1\[ \times \]\[\dfrac{{364}}{{365}}\]
\[ = \]\[\dfrac{{364}}{{365}}\]
Therefore, the probability of two of them having the same birthday
\[ = \] 1\[ - \] probability (two of them having different birthdays)
\[ = \] 1 \[ - \]\[\dfrac{{364}}{{365}}\]
\[ = \]\[\dfrac{{365 - 364}}{{365}}\]
\[ = \] \[\dfrac{1}{{365}}\]
Hence, probability that both of them have,
1.same birthdays is \[\dfrac{1}{{365}}\].
2.different birthdays is \[\dfrac{{364}}{{365}}\].

Note:
It is very important to read the question carefully and find the answer which is easy to find first rather than the order. Many students try to find the probability of them having the same birthday and get stuck there as they cannot relate their concept with basic formulas they have used in probability. It can be solved also by finding the probability of both of them having the same birthday and then we need to just subtract it from 1 to get the probability that both of them have different birthdays. To find the probability of both of them having same birthday rather than the formulas we use just a basic understanding that in this case P(A) \[ = \]\[\dfrac{{365}}{{365}}\] like before but P(B) when they have same birthday is \[\dfrac{1}{{365}}\] as the favourable outcome is 1 because we need that particular day when Sourte’s has been born.