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# If there are a certain number of students in a classroom

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Manager
Joined: 25 Sep 2018
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GMAT 1: 640 Q47 V30
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If there are a certain number of students in a classroom  [#permalink]

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14 Oct 2018, 00:16
1
4
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Difficulty:

65% (hard)

Question Stats:

41% (01:24) correct 59% (01:26) wrong based on 37 sessions

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If there are a certain number of students in a classroom, what is the probability that at least 3 of them have birthdays in the same month?
1) There are no students born in the month of December.
2) There are 25 students in the classroom.

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Re: If there are a certain number of students in a classroom  [#permalink]

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14 Oct 2018, 03:37
Abhi077 wrote:
If there are a certain number of students in a classroom, what is the probability that at least 3 of them have birthdays in the same month?
1) There are no students born in the month of December.
2) There are 25 students in the classroom.

As this is a DS question, all we need to do is figure out if we could calculate the answer should we want to.
Especially in questions like this, where actual calculation is complicated, it helps to keep this in mind.

In particular, we need to calculate "the number of ways to pick birthdays for students so that at least 3 students have birthdays in the same month" and divide it by "the total number of ways to pick birthdays for students". Since we know the number of months/days in a year, and assuming birthdays are evenly distributed, all we really need to know to calculate is the total number of students.
This is what we'll look for and is a Logical approach to solution.
As (1) does not give us the total number of students and (2) does, (2) is sufficient.

If this sort of logic is confusing for you, you can instead try small numbers to help make things more concrete.
This is an Alternative approach.

(1) Say we have only 2 students or only 1 student in the classroom. Then the probability of having 3 together is 0. On the other hand, say we have 100, or 500, or some other really large number of students. Then the probability is definitely positive, meaning different from 0.
Insufficient.

(2) Say we take our 25 students and put them into different months. Since putting 2 students in every month uses up only 2*12=24 students, then the next student must be assigned to a month that already has 2 students in it. Therefore we are guaranteed to have at least one month with 3 students' birthdays and our probability is 1.
Sufficient.
(For those interested, the logic in (2) is called the 'pigeonhole principle'.)
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Re: If there are a certain number of students in a classroom  [#permalink]

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15 Oct 2018, 05:07
Solution: Question Stem Analysis: We must determine the probability that at least 3 of the students have a birthday in the same month.
Statement One Alone: There are no students born in the month of December.
Thus, the students' birthdays are all in the other 11 months, but without more information about the number of students in the classroom, we cannot answer the question. Statement one alone is not sufficient.
Eliminate Answer Choices: A and D Statement Two Alone: There are 25 students in the classroom.
Statement Two Alone: There are 25 students in the classroom.
We can approach this by trying to spread out the birthdays as much as possible. Let's place the first 12 students, one at a time, in the 12 different months. After doing so, each month has a student with a birthday in it. This leaves us with 13 students. Take 12 of these students and place them one at a time in those same 12 months. After doing so, each month has 2 students with the same birthday in it. There is one remaining student whom we have not yet placed. We can place the last student in any of the 12 months. Regardless of which month this student goes to, there will be at least 3 students with a birthday in the same month. Thus the probability is 100 percent. Answer B
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Re: If there are a certain number of students in a classroom   [#permalink] 15 Oct 2018, 05:07
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