Question

Assume the year to be ordinary consisting of 365 days. What is the probability that in a group of 3 people, at least 2 will have same date of birth,
A. \[1-\dfrac{364\times 363}{{{\left( 365 \right)}^{2}}}\]
B. \[1-\dfrac{2\times 363}{{{\left( 365 \right)}^{2}}}\]
C. \[1-\dfrac{364\times 2}{{{\left( 365 \right)}^{2}}}\]
D. None of the above

Answer 1

Hint: In this problem, we have to find the probability that in a group of 3 people, at least 2 will have the same date of birth in an ordinary year consisting of 365 days. Here we have to find the people who do not have the same date of birth and subtract with the total probability i.e. 1. We are given that there are three people, the first person who does not have the same date of birth will have 356 ways among 365 days and the second person does not have the same date, so we have to exclude the date selected by the first person and process repeats for the third person. We can then subtract to get the answer.

Complete step by step answer:
Here we have to find the probability that in a group of 3 people, at least 2 will have the same date of birth in an ordinary year consisting of 365 days.
Here we have to find the people who do not have the same date of birth and subtract it with the total probability i.e. 1.
We are given that there are three people, the first person who does not have the same date of birth will have 356 ways among 365 days.
\[= \dfrac{365}{365}\]
 Similarly, the second person does not have the same date, so we have to exclude the date selected by the first person.
\[= \dfrac{364}{365}\]
Similarly, for third person
\[= \dfrac{363}{365}\]
We can now subtract the persons with different date of birth from the total probability,
Probability that at least 2 will have same date,
\[\begin{align}
  & = 1-\dfrac{365}{365}\times \dfrac{364}{365}\times \dfrac{363}{365} \\
 & = 1-\dfrac{364\times 363}{{{\left( 365 \right)}^{2}}} \\
\end{align}\]

So, the correct answer is “Option A”.

Note: We should always remember that, the total probability is always 1, so here we have subtracted the persons with different dates of birth to the total probability to get the person with the same date of birth. We should solve the given data up to the given options in the question.