Answer
Verified
396.6k+ views
Hint: At first , excluding leap years there are 365 different birthdays possible , so in a group of N people the person can have birthdays in ${\left( {365} \right)^N}$different ways. This can be denoted by n . Next now we can calculate the number of different ways that no people have the same birthday. We get that the number of different ways is .$m = 365*364*363*...*(365 - N + 1)$.
Complete step-by-step answer:
Step 1 :
We are asked to find the probability that in a group of ‘N’ people , at least two of them have the same birthday.
Firstly , let's find the probability that no two persons have the same birthday and subtract it from 1 as the total probability of a success event is 1 .
Step 2:
Excluding leap years there are 365 different birthdays possible
So , any person can have any one of the 365 days of the year as a birthday
Same way the second person may also have any one of the 365 days of the year as a birthday and so on.
Hence in a group of N people , there are ${\left( {365} \right)^N}$possible combination of birthdays
So now let n = ${\left( {365} \right)^N}$
Step 3 :
Now assuming that no two people have their birthday on the same day
The first person can have any one of the 365 days as his birthday
So the second person will have his birthday in any one of the 364 days
And the third person will have his birthday in any one of the 363 days and so on
From this , we can get that the Nth person may have his birthday in any one of the ( 365 – N +1 ) days
The number of ways that all N people can have different birthdays is then
$m = 365*364*363*...*(365 - N + 1)$
Therefore , the probability that no two birthdays coincide is given by $\dfrac{m}{n}$
$ \Rightarrow \dfrac{m}{n} = \dfrac{{365*364*363*...*(365 - N + 1)}}{{{{\left( {365} \right)}^N}}}$
Step 4 :
Probability that at least two person will have the same birthday = 1 – (probability that no two birthdays coincide)
$
\Rightarrow 1 - \dfrac{{365*364*363*...*(365 - N + 1)}}{{{{\left( {365} \right)}^N}}} \\
\\
$
The above expression gives the probability that at least two people will have the same birthday.
Note: The assumptions that a year has 365 days and that all days are equally likely to be the birthday of a random individual are false, because one year in four has 366 days and because birth dates are not distributed uniformly throughout the year. Moreover, if one attempts to apply this result to an actual group of individuals, it is necessary to ask what it means for these to be “randomly selected.” It would naturally be unreasonable to apply it to a group known to contain twins.
Complete step-by-step answer:
Step 1 :
We are asked to find the probability that in a group of ‘N’ people , at least two of them have the same birthday.
Firstly , let's find the probability that no two persons have the same birthday and subtract it from 1 as the total probability of a success event is 1 .
Step 2:
Excluding leap years there are 365 different birthdays possible
So , any person can have any one of the 365 days of the year as a birthday
Same way the second person may also have any one of the 365 days of the year as a birthday and so on.
Hence in a group of N people , there are ${\left( {365} \right)^N}$possible combination of birthdays
So now let n = ${\left( {365} \right)^N}$
Step 3 :
Now assuming that no two people have their birthday on the same day
The first person can have any one of the 365 days as his birthday
So the second person will have his birthday in any one of the 364 days
And the third person will have his birthday in any one of the 363 days and so on
From this , we can get that the Nth person may have his birthday in any one of the ( 365 – N +1 ) days
The number of ways that all N people can have different birthdays is then
$m = 365*364*363*...*(365 - N + 1)$
Therefore , the probability that no two birthdays coincide is given by $\dfrac{m}{n}$
$ \Rightarrow \dfrac{m}{n} = \dfrac{{365*364*363*...*(365 - N + 1)}}{{{{\left( {365} \right)}^N}}}$
Step 4 :
Probability that at least two person will have the same birthday = 1 – (probability that no two birthdays coincide)
$
\Rightarrow 1 - \dfrac{{365*364*363*...*(365 - N + 1)}}{{{{\left( {365} \right)}^N}}} \\
\\
$
The above expression gives the probability that at least two people will have the same birthday.
Note: The assumptions that a year has 365 days and that all days are equally likely to be the birthday of a random individual are false, because one year in four has 366 days and because birth dates are not distributed uniformly throughout the year. Moreover, if one attempts to apply this result to an actual group of individuals, it is necessary to ask what it means for these to be “randomly selected.” It would naturally be unreasonable to apply it to a group known to contain twins.
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 Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Two charges are placed at a certain distance apart class 12 physics CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is BLO What is the full form of BLO class 8 social science CBSE