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P(they all were born in the same month ) = 1 / 12^12 = ???
P(they all were born in different months ) = 12! / 12^12 = ???
P(exactly six of them were born in the same month) =
= 11 * 10 * 9 * 8 * 7 * 6 / 12^12

The probability that each prisoner was born in January, for example
1/12*1/12*1/12....=1/12^12
If we take all months 12*1/12^12
The probability that all were born in different months
12!/12^12
The probability that exactly six of them were born in the same month
12C6/12^12

The probability that each prisoner was born in January, for example 1/12*1/12*1/12....=1/12^12 If we take all months 12*1/12^12 The probability that all were born in different months 12!/12^12 The probability that exactly six of them were born in the same month 12C6/12^12

(1) correct
(2) correct
(3) doubt

any 6 are taken (12C6). they were born in the same month (12). other six people have 11*10*9*8*7*6 chances to be born in different months.

I got the first 2 right but I think that the 3rd question should be worded as: The probability that exactly six of them were born in the same month
and that the 5 others are all born in different months.
Only then will Stoylar's answer's 12C6*12*11*10*9*8*7*6/12^12 will be good. Because, what if 6 have the same birth months but 2, 3 , 4 or 5 others also have another same birth month? I think worded as I said it, then Stoylar's answer would be right. Any thoughts? _________________

I think the 1st one should be 12/(12^12) because there are 12 possible months for everyone to be born on the same month. 1/(12^12) implies that there is only one possible way, but everyone can be born in Jan, Feb, Mar...etc all the way to Dec.

For example, if there are only 3 months in a year (Jan , Feb, Mar) and three people, then the total possible combinations are 3/(3^3).

P1 P2 P3
Jan Jan Jan
Feb Feb Feb
Mar Mar Mar
-----------------
Jan Feb Mar
Jan Mar Feb
Feb Jan Mar
Feb Mar Jan
Mar Jan Feb
Mar Feb Jan
21 more ....

I think the last one should be 12C6*12*11*11*11*11*11*11/12^12. Six can be born in the same month and the other six can be born in the same months too...just not the same month as the first six.

Calnhob, I think your answer should be good. 12C6*12 for 6 having the same birth months, then 11^6 for the other 6 having diff. or same birth months all being accounted for in 11^6. Thus, the final answer, as you said should really be (12C6*11^6) / 12^12. Do you agree that Stoylar's answer represents The probability that exactly six of them were born in the same month and that the 5 others are all born in different months.? _________________