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In a family of four, what is the probability that no two people have the same birthdays in the same month.(Assuming that all the months have equal probability)

Firstly I understood the question to be similar to the one that asks for the probability of no two people having the sam birthdate. So I tried that P(of no two people sharing the same Bday in same month)= 1-P( two people have the same Bday)

1-4C2/30 (Assuming that every month has 30 days. =1-0.2 =0.8.

Am I on the right track?

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  • $\begingroup$ I don't understand. Are you simply asking what is the probability that no two family members share a birthday? Or that no two family members share a birthmonth? If the former, why do you specify months? $\endgroup$ – Xander Henderson Dec 19 '17 at 1:03
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    $\begingroup$ Hard to follow. what does the $30$ have to do with anything? Order them (by age, say). The probability that the first two have different birthmonths is $\frac {11}{12}$. The probability that the third has yet a different birthmonth is then $\frac {10}{12}, and $\frac 9{12}$ for the fourth. Thus.... $\endgroup$ – lulu Dec 19 '17 at 1:27
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As Xander Henderson said, if the question is just 'what is the probability that two people have the same birthday' then individual months is not relevant- though the year, whether a leap year or not, can.

There are 365 days in a non-leap year. Person "A" has some birthday. In order that person "B" has the same birthday, his birthday must be that 1 day out of 365. The probability of that is 1/365.

There are 365 days in a leap year. Person "A" has some birthday. In order that person "B" has the same birthday, his birthday must be that 1 day out of 366. The probability of that is 1/366.

To get the "overall" probability, use the fact that 1/4 of all years are leap years, 3/4 are not*. So the probability two people share the same birthday is (3/4)(1/365)+ (1/4)(1/366).

  • Strictly speaking, this is not exactly true. In the Gregorian calendar, Any year divisible by 100 but not divisible by 400, is NOT a leap year. 1800 and 1900, though divisible by 4 were not leap years because they are divisible by 100. The year 2000 was a leap year. It is divisible by 100 but also by 400.
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