[Math] What’s the probability that 2 of 4 people were born in the same month

combinatoricsprobability

I have this problem to solve:

"Calculate what's the probability that in the four-person family at least 2 people were born in the same month".

I know that I can calculate opposite event in very simple way but I'd like to know what's wrong with my reasoning.

We need to at least 2 people so we can make 3 cases:

  1. Two people were born at the same month.
  2. Three people were born at the
    same month.
  3. Four people were born at the same month.

So in the first case firstly we choose 2 people of 4, which were born at the same month, then multiple by 12 possible months and then by 11 and 10 i.e. by possible months for respectively third and fourth person. In other cases I do it analogously.

So the result is:
$\binom{4}{2}*12*11*10 + \binom{4}{3}*12*11+\binom{4}{4}*12 = 8640$
But the result should be 8856.

Edit:
I've got it. As you said, there are four cases. I've wrote them symbolically.

  1. X X Y Z
  2. X X Y Y
  3. X X X Y
  4. X X X X

First, third and fourth case I've described before. In the second case we need to first chose two people then a month for them and then a month for two other people. But in this case we must notice that the results double each other.For example in first step we chose person '1' and '3' and a month '6' for them. For person '2' and '4' we chose month '5'. So we have the result: 6 5 6 5. But we can alson first chose person '2' and '4' and a month '5' for them and for person '1' and '3' chose month '6'. So we have 6 5 6 5 again.

So final result is:
$\binom{4}{2}*12*11*10 + \binom{4}{2}*12*11*\frac 12 + \binom{4}{3}*12*11+\binom{4}{4}*12 = 8856$

Best Answer

The four cases you need to consider are:

  • Two people born on the same month, two on different months. (three months)
  • Two lots of two people born on the same month, for two different months.
  • Three people born on the same month, one on a different month.
  • Four people born on the same month.

You missed the case in bold.

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