Probability that the birthdays of 6 different persons will fall in exa

Question

Probability that the birthdays of 6 different persons will fall in exactly 2 calendar months is

A. 341/12^6
B. 352/12^6
C. 352/12^5
D. 341/12^5
E. 371/12^6

fluke · Accepted Answer

Sol:

Assuming that any person having a birthday on any month to be  \frac{1}{12} . In reality; probability of a person having a birthday on Feb(28 days) is less than having a birthday on January(31 days).

C^{12}_2*\frac{2^6-2}{12^6}=\frac{341}{12^5}

C^{12}_2   - Number of ways to choose 2 months out of 12 
 2^6-2   - 2 favorable choices(months) available for every person excluding 000000,111111(All Birthdays in the same month) 
 12^6    - Every person has 12 Total possible months

Ans: "D"

fluke · Answer

6 people have two favorable choices each; Let's say the choices were 0(JAN),1(FEB) for simplicity just as it would be for head and tail.

Thus, total number of combinations would be: 2^6.

000000
000001
000010
000011
.
.
111111

Total=2^6

As per the condition that all 6 must have EXACTLY two calender months as their birthdays, at least two people should have different birthdays. They all can't have 000000 i.e. JAN,JAN,JAN,JAN,JAN,JAN or 1,1,1,1,1,1 i.e. FEB,FEB,FEB,FEB,FEB,FEB.

So, Favorable possibilities = 2^6-2=62.

subhashghosh · Answer

@fluke, please explain this :

000000,111111(All Birthdays in the same month)

KarishmaB · Answer

Another slightly different way of approaching this (and a little more complicated than fluke's - which I think is a very appropriate solution. Nevertheless, it might be useful for people who have done permutation combination before). 
Select 2 months and then no of people for each month. You select two months in 12C2 ways. Say you get J and F.
Then you say that J could have 1 person's bday, 2 people's bday, 3 people's bday etc

J - 1/2/3/4/5 (You select 1 or 2 or 3 or 4 or 5 people for J)
F - 5/4/3/2/1 (Corresponding to no of people selected for J, remaining people go in F)

The distribution can be done in  6C1 + 6C2 + 6C3 + 6C4 + 6C5 = 2^6 - 2 = 62  ways (If you recognize it, great, else just calculate)

So probability  = 12C2 * 62/ 12^6 = 341/12^5

deepakh88 · Answer

My understanding of your solution goes this way.Let me know if I'm wrong.
1. Selecting any two months out of the 12 available can be done in 12C2 ways.
2. The 6 days can be distributed in the 2 available months in the following ways

M1 M2
0 6
1 5
2 4
3 3
4 2
5 1
6 0

which essentially is 6C0 + 6C1 + 6C2 + 6C3 + 6C4 + 6C5 + 6C6,
which is of the form 
(x+a)^6 = x^6 + x^5 6C1 a^1 + x^4 6C2 a^2 + x^3 6C3 a^3 + . . + a^6.
Substituting 1 for 'x' and 'a' we arrive at (1+1)^6 = 1 + 6C1 + 6C2 + 6C3 + 6C4 + 6C5 + 1
2^6 = 2 + 6C1 + 6C2 + 6C3 + 6C4 + 6C5
i.e, 2^6-2 = 6C1 + 6C2 + 6C3 + 6C4 + 6C5 = 62
Here, we subtract 2 because we are not to take into account either cases where all 6 birthdays fall in M1 or M2

Thus the probability becomes 12C2 * 62/ 12^6 = 341/12^5

gmat1220 · Answer

Exactly as the binomial dist you have written! That is basically weeding out all the cases which have "all" 6 birthdays in one month i.e. 6C6 and 6C0

KarishmaB · Answer

Yes, that's correct. Many of us have used binomial theorem extensively and know that:
 nC0 + nC1 + nC2 + ..... nCn = 2^n  (an application of binomial as described by you)
Hence, it can be done this way.

On the other hand, you could obviously say that each of the six people have 2 options - either his bday falls in J or in F. So those are a total of  2*2*2*2*2*2 = 2^6  ways. Out of those, we need to remove 2 ways: All bdays fall in J and all bdays fall in F. Hence it can be done in  (2^6 - 2)  ways.

shashankism · Answer

Really mind boggling question, You can't solve it in one go unless and until you have solved such question before..
 fluke  &  VeritasPrepKarishma  very nice solution... 
It seems easy once we read the solution but it's a real tough question..

KarishmaB · Answer

Responding to a pm:
 
12C2 = 12*11/2

\frac{12C2 * 62}{12^6} = \frac{12*11*62}{2*12^6} = \frac{11*31}{12^5} = \frac{341}{12^5}

Appus · Answer

How did n(s) become 12*6

Posted from my mobile device

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Probability that the birthdays of 6 different persons will fall in exa

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