Question

A and B are friends. Ignoring the leap year, find the probability that both friends will have different birthdays is $\dfrac{m}{365}$. What is the value of m?

Answer 1

In this question, we need to find the probability of them having their birthday on the same day by finding the total number of ways and favourable ways then get the probability using the formula $P=\dfrac{m}{n}$. Here, we know that there are 365 days in a year and assume that the probability of them having birthday on the same day as $P\left( A \right)$and probability of them having birthday on different days as $P\left( B \right)$. Now, subtracting $P\left( A \right)$ from 1 gives $P\left( B \right)$.

Complete step by step answer:
PROBABILITY: If there are n elementary events associated with a random experiment and m of them are favourable to an event A, then the probability of happening or occurrence of A, denoted by P(A), is given by
$P\left( A \right)=\dfrac{m}{n}=\dfrac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$
Let the event of both of them having birthday on the same day as A
Now, there are 365 days in total in a non leap year
Here, the number of total outcomes possible for them to have birthday on any day is given by
$\Rightarrow 365\times 365$
Now, the favourable outcomes for them to have birthday on the same day can be any of the day in 365 days which is given by
$\Rightarrow 365$
Now, on comparing these values with formula of probability we get,
$m=365,n=365\times 365$
As we already know the formula to find the probability is given by
$P=\dfrac{m}{n}$
Now, on substituting the respective values in the above formula we get,
$\Rightarrow P\left( A \right)=\dfrac{365}{365\times 365}$
Now, on simplifying this further we get,
$\therefore P\left( A \right)=\dfrac{1}{365}$
Now, let us assume that the event of them having their birthdays on different as B
Here, the events A and B are mutually exclusive events so we get,
$\Rightarrow P\left( A \right)+P\left( B \right)=1$
Now, on substituting the respective values we get,
$\Rightarrow \dfrac{1}{365}+P\left( B \right)=1$
Now, on rearranging the terms we get,
$\Rightarrow P\left( B \right)=1-\dfrac{1}{365}$
Now, on further simplification we get,
$\therefore P\left( B \right)=\dfrac{364}{365}$
Now, on comparing this value with the given value in the question we get,
$\Rightarrow \dfrac{m}{365}=\dfrac{364}{365}$
Now, on cancelling the common terms we get,
$\therefore m=364$

Note:
Instead of finding the probability of them having birthdays on the same day and then subtracting it from 1 we can also find it by finding the total number of ways which will be the same and the number of favourable ways can be found by using the permutation formula by arranging their birthdays in any 2 days out of 365 days.
It is important to note that the sum of the probability of them having their birthdays on the same day and different day will be 1 because they either can have on same day or different and no other way. This method will be a bit easy to calculate.