Question

Sita and Geta are friends, what is the probability that both will have different birthdays (ignoring a leap year)
A. \[\dfrac{1}{{365}}\]
B. \[\dfrac{1}{{364}}\]
C. \[\dfrac{{364}}{{365}}\]
D. None of these

Answer 1

Hint: To solve this question let’s assume the birth date and month of sita. Then find the probability of getting the birthday of geta on the same date and month. Then we get the probability of getting a same-day birthday. But we have to find the probability that both will have different birthdays. So for finding the probability that both will have different birthdays we will subtract the probability of getting the same birthday from \[1\].

Complete step-by-step answer:
To find,
The probability of getting the different birthday
\[Probability{\text{ }}of{\text{ }}getting{\text{ }}different{\text{ }}birthday\; = 1 - probability{\text{ }}of{\text{ }}getting{\text{ }}the{\text{ }}same{\text{ }}birthday\]So, now we are going to find the probability of getting the same birthday
For finding the probability of getting the same birthday
Lets, assume the birthdate of sita
Now, what is the chances are possible out of \[365\] to get the same birthday?
Favorable number of outcomes \[ = 1\]
Total number of outcomes \[ = 365\]
Probability of getting same birthday is the ratio of favorable number of outcomes to the total number of outcomes.
\[Probability{\text{ }}of{\text{ }}getting{\text{ }}same{\text{ }}birthday = \dfrac{{favorable{\text{ }}number{\text{ }}of{\text{ }}outcomes}}{{total{\text{ }}number{\text{ }}of{\text{ }}outcomes}}\]
\[Probability{\text{ }}of{\text{ }}getting{\text{ }}same{\text{ }}birthday = \dfrac{1}{{365}}\]
\[Probability{\text{ }}of{\text{ }}getting{\text{ }}different{\text{ }}birthday\; = 1 - probability{\text{ }}of{\text{ }}getting{\text{ }}the{\text{ }}same{\text{ }}birthday\]
On putting the value of probability of getting same birthday
\[Probability{\text{ }}of{\text{ }}getting{\text{ }}different{\text{ }}birthday\; = 1 - \dfrac{1}{{365}}\]
On taking the LCM in denominator
\[Probability{\text{ }}of{\text{ }}getting{\text{ }}different{\text{ }}birthday\; = \dfrac{{365 - 1}}{{365}}\]
On further solving
\[Probability{\text{ }}of{\text{ }}getting{\text{ }}different{\text{ }}birthday\; = \dfrac{{364}}{{365}}\]
Final answer:
Probability of getting different birthday is
\[ \Rightarrow p(e) = \dfrac{{364}}{{365}}\]

So, the correct answer is “Option C”.

Note: To solve this type of question, if we think from the negative direction the solution of the question is much easier. In this particular case if we are asked to find the probability of getting a different birthday but we are finding the probability of getting the same birthday and then subtract from \[1\] to get the Probability of getting a different birthday.