Question

Savitha and Hamida are friends, what is the probability that both will have (i) different birthdays. (ii) the same birthday (ignoring a leap year).

Answer 1

This problem deals with probability, which is given by the ratio of the favorable outcome to the total number of outcomes. Here we have to find the probability that both sisters have their birthdays on the same day of the year, ignoring the leap year. A normal year has 365 days, whereas the leap year has 366 days.

Complete step-by-step solution:
First we will calculate the probability that both friends have their birthdays on the same day
(ii) Probability both have the same birthday.
Here given to ignore the leap year.
Here probability of a favorable event is given by the ratio of the favorable outcome to the total no. of outcomes.
The probability that Savitha has her birthday on a particular day, is given by:
$\Rightarrow \dfrac{1}{{365}}$
Now the probability that Hamida has her birthday on the same particular day as Savitha, is given by:
$\Rightarrow \dfrac{1}{{365}}$
Hence the probability that both the friends have their birthdays on the same day is given by the product of the probability that Savitha has her birthday on one day and the probability that Hamida has her on the same day as Hamida. Which is expressed mathematically below:
$\Rightarrow \dfrac{1}{{\left( {365} \right)}} \times \dfrac{1}{{\left( {365} \right)}}$
$\Rightarrow \dfrac{1}{{{{\left( {365} \right)}^2}}}$
Hence the probability that both the friends have their birthday on same day is $\dfrac{1}{{{{\left( {365} \right)}^2}}}$
Now calculating the probability that both friends have their birthdays on different days.
(i)Probability both have different birthdays.
$\Rightarrow 1 - \dfrac{1}{{{{\left( {365} \right)}^2}}} = \dfrac{{{{\left( {365} \right)}^2} - 1}}{{{{\left( {365} \right)}^2}}}$
Hence the probability that both the friends have different birthdays is given by: $1 - \dfrac{1}{{{{\left( {365} \right)}^2}}}$

Note: This problem can be done in another method as well, where the other method is described here. The total no. of outcomes in the event is $365 \times 365$, as these are total no. of chances for both the sisters combined. The probability of the favorable outcome is 1 here, as both need to have their birthdays on the same day. Hence applying the formula of probability of the favorable event gives the same final answer.