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there are 27 students in a class.what is the prob that atleast 3 of them have their birthday in the same month?


the solution suggested for the question the booklet says avg birthday prob = 27/12 =2.something

so, according to pigeonhole principle, the required probability is 1.

can someone please explain me this.

in Probability by (24 points) | 45 views

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This is a classic pigeonhole principle application.

There are 12 months in a year. So assume 12 rooms, 1 room for each month. You want to put 27 students into these rooms, uniformly. How will you do that ?

  • 1st round, 1 student into each room. Now each room contains 1 student, 15 students remain.
  • 2nd round, 1 student again into each room. Now each room contains 2 students, 3 students remain.
  • 3rd round, you can send the 3 remaining students into any of the rooms. After this, It is guaranteed that some room will have 3 students (if not more).

Hence you can conclude that there will definitely be a room with 3 students. With this analogy, you can conclude that in a group of 27 students, there will surely be at least 3 students whose birthday is in the same month.

And since the event is guaranteed to occur, probability is obviously 1.

by (1.9k points)
how did you decide to distribute the students uniformly???

why can't it be the case that all the students have their birthday's in the same month.

Of course you could have all the students have the birthday in the same month. That directly satisfies the condition question right ?

atleast 3 of them have their birthday in the same month

Question asked for at least 3, and you gave all of them..

The point of distributing uniformly, is to ensure the minimum amount required to guarantee that the condition is fulfilled. Please read the pigeon hole principle from a reference book to understand this more.


@shashin As always great answer 

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