Question

The probability that in a group of $n$ people, at least two of them will have the same date of birth of a non leap year (Assuming a year is having 365 days).
A) $1 - \dfrac{{{}^{365}{P_n}}}{{{{\left( {365} \right)}^n}}}$
B) $\dfrac{{{}^{365}{P_n}}}{{{{\left( {365} \right)}^n}}}$
C) $\dfrac{1}{{{{\left( {365} \right)}^n}}}$
D) $\dfrac{{365 \times 365}}{{{{\left( {365} \right)}^n}}}$

Answer 1

Hint:
Here, we will first find the probability of people having different birth dates. Then we will use the formula of complementary events and subtract the obtained probability from 1 to get the required probability. Probability is defined as the certainty of the occurrence of an event. The sum of all the probabilities is equal to 1.

Formula used:
We will use the below formula $P\left( A \right) = 1 - P\left( {\bar A} \right)$, where, $P\left( A \right)$ is Probability of at least two people having same birth date and $P\left( {\bar A} \right)$ is the probability of people having different birth date.

Complete step by step solution:
As it is given the number of days in a year is 365 as it is a non-leap year. So, each one of the people in the group can have their birthday in 365 ways. Therefore, $n$ people can have their birthday in ${\left( {365} \right)^n}$ ways.
Now, the number of ways in which all have different birthdays $ = {}^{365}{P_n}$
So, the probability of people having a different birth date is given as
$P \left( {\bar A} \right) = \dfrac{{{}^{365}{P_n}}}{{{{\left( {365} \right)}^n}}}$ ……………………………………$\left( 1 \right)$
So now we will get the probability of people having birth date same by subtracting equation $\left( 1 \right)$ from total probability i.e.$1$ , we get
$P\left( A \right) = 1 - P\left( {\bar A} \right) \\
   \Rightarrow P\left( A \right) = 1 - \dfrac{{{}^{365}{P_n}}}{{{{\left( {365} \right)}^n}}} \\ $
So probability of at least two people having same birth date is $1 - \dfrac{{{}^{365}{P_n}}}{{{{\left( {365} \right)}^n}}}$.

Hence, option (A) is correct.

Note:
Here,$A$ and $\bar A$ are the only two possibilities here and they also are mutually exclusive events, so we can easily find the required probability by using the formula for complementary events. Probability of an event can never be less than 0 or more than 1. An event is called a sure event if its probability is 1, whereas if the probability of event is 0, the event will never occur.