Answer
Verified
394.2k+ views
Hint: Calculate the total number of possible outcomes when all 10 persons are born in June. We will then calculate the number of possible ways if all the 10 persons have birthdays on different days. Then, we will subtract the cases when all the 10 persons have birthdays on different days from the total number of possible cases.
Complete step-by-step answer:
Each person will have a birthday on some date in June. There are 30 days in June. So, the total number of possible outcomes will be ${30^{10}}$.
We now have to find out if at least two people will have birthdays on the same date.
We first find the number of possible ways if all the 10 persons have birthdays on different days.
Suppose all the 10 persons have birthdays on different days. Then, the total number of cases will be $^{30}{C_{10}}$.
To find the probability that at least two have the same birthday, we will subtract the cases when all the 10 persons have birthdays on different days from the total number of possible cases.
Hence, total number of possibilities if at least two have the same birthday is ${30^{10}}{ - ^{30}}{C_{10}}$
Now, we will calculate the probability using the formula, $\dfrac{{{\text{number of favourable outcomes}}}}{{{\text{total number of possible outcomes}}}}$
Hence, the required probability is $\dfrac{{{{30}^{10}}{ - ^{30}}{C_{10}}}}{{{{30}^{10}}}}$
Hence, option D is correct.
Note: If we want to find that at least two have the same birthday, then we have to include all the possibilities of 2 persons, 3 persons, and so on till 10 persons have birthday on the same day. This will be a very lengthy process. Hence, we will first find when no two persons have birthdays on the same date and subtract it from all the possible outcomes.
Complete step-by-step answer:
Each person will have a birthday on some date in June. There are 30 days in June. So, the total number of possible outcomes will be ${30^{10}}$.
We now have to find out if at least two people will have birthdays on the same date.
We first find the number of possible ways if all the 10 persons have birthdays on different days.
Suppose all the 10 persons have birthdays on different days. Then, the total number of cases will be $^{30}{C_{10}}$.
To find the probability that at least two have the same birthday, we will subtract the cases when all the 10 persons have birthdays on different days from the total number of possible cases.
Hence, total number of possibilities if at least two have the same birthday is ${30^{10}}{ - ^{30}}{C_{10}}$
Now, we will calculate the probability using the formula, $\dfrac{{{\text{number of favourable outcomes}}}}{{{\text{total number of possible outcomes}}}}$
Hence, the required probability is $\dfrac{{{{30}^{10}}{ - ^{30}}{C_{10}}}}{{{{30}^{10}}}}$
Hence, option D is correct.
Note: If we want to find that at least two have the same birthday, then we have to include all the possibilities of 2 persons, 3 persons, and so on till 10 persons have birthday on the same day. This will be a very lengthy process. Hence, we will first find when no two persons have birthdays on the same date and subtract it from all the possible outcomes.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE