Question

Ashmita and Shreya are sisters, what is the probability that both have birthday on 14 September (ignoring leap year)
A) $ \dfrac{1}{{30}} $
B) $ \dfrac{2}{{365}} $
C) $ \dfrac{1}{{366}} $
D) $ \dfrac{1}{{{{\left( {365} \right)}^2}}} $

Answer 1

Hint: To find the required probability, we will find the favorable and total outcomes for the given event and then substitute it in the formula for probability.
A year which does not leap has 365 days in total.
Formula to be used:
 $ P = \dfrac{f}{T} $ where, P is the probability, f is favorable outcomes and T is the total number of outcomes.

Complete step-by-step answer:
There are 365 days in a year which is not a leap. Ashmita can have her birthday on one out of the total 365 days , similarly Shreya can have her birthday on one out of the total 365 days.
So, the total number of possible outcomes of their birthdays are:
 $ \Rightarrow 365 \times 365 = {\left( {365} \right)^2} $
If they both have their birthday on the same day i.e. 14 September, the favorable outcome for such an event will be 1.
So the required probability is given as:
 $ \Rightarrow P = \dfrac{f}{T} $ here,
Favorable outcomes (f) = 1
Total outcomes (T) = $ {\left( {365} \right)^2} $
Substituting, we get:
 $ \Rightarrow P = \dfrac{1}{{{{\left( {365} \right)}^2}}} $
Therefore, the probability that both Ashmita and Shreya have birthday on 14 September is $ \dfrac{1}{{{{\left( {365} \right)}^2}}} $
So, the correct answer is “Option D”.

Note: It has been specifically mentioned in the question to ignore leap year. Leap year has 366 days instead of 365, but when ignored we directly consider the number of days in a year as 365. We could also have calculated the value of $ {\left( {365} \right)^2} $ , but as we had to answer amongst the options, we left it as it is and it is also easier leaving like this than to calculate this square.