Probability 5 people same birthday day of week

[Jon R]

Hi,
I hope you please might be able to help.
I’m not a student, just a son and dad with a family probability question we’d love to know the answer to, please!
Our 5 family birth dates are:
2nd March 1931
8th June 1932
5th January 1967
6th October 2001
1st December 2004
As you will see, we were not born on the same days of the week.
However, EVERY year, regardless of leap years, whatever day of the week our daughter’s birthday on 6th October falls on, the other 4 members of our family then have their birthdays fall on the same day of the week up until the birthday the following year in June.
We realise this is due to our birthdays being always a certain number of days apart, but to us the probability of this scenario happening seems very low.
We’d love for one of you maths experts to tell us the probability....perhaps the scenario is not as unlikely as we like to think!
Many thanks in anticipation of anyone who feels able to help.
Jon
England

 

[Romsek]

First person picks a birthdate/day with probability 1

6 days occur 52 times a year.
1 day occurs 53 times a year.

The other four now have either 52 or 53 choices

[MATH]P[\text{all have same birthdate day}] = \dfrac 1 7 \left(\dfrac{53}{365}\right)^4 + \dfrac 6 7 \left(\dfrac{52}{365}\right)^4 \approx 0.00042 = 0.042\%[/MATH]
Taking leap year into account complicates this a fair bit.

0.042% is pretty rare in my book.

 

[Dr.Peterson]

Hi,
I hope you please might be able to help.
I’m not a student, just a son and dad with a family probability question we’d love to know the answer to, please!
Our 5 family birth dates are:
2nd March 1931
8th June 1932
5th January 1967
6th October 2001
1st December 2004
As you will see, we were not born on the same days of the week.
However, EVERY year, regardless of leap years, whatever day of the week our daughter’s birthday on 6th October falls on, the other 4 members of our family then have their birthdays fall on the same day of the week up until the birthday the following year in June.
We realise this is due to our birthdays being always a certain number of days apart, but to us the probability of this scenario happening seems very low.
We’d love for one of you maths experts to tell us the probability....perhaps the scenario is not as unlikely as we like to think!
Many thanks in anticipation of anyone who feels able to help.
Jon
England

Something seems a little wrong there; the January and March/June dates are separated by leap day, so what you describe should not be true when that is a leap year.

My own family is similar; four of the five of us have birthdays (9/9, 9/16, 10/7, 1/6) that always fall on the same day of the week. Leap day does not intervene, making this possible.

In any case, events with low probability occur all the time! Taking 0.042% as correct, there are something like 20 million families in the U.K, according to the first hit I get in Google, so there might be about 8400 families like yours.

So I don't think there's any fraud involved in your births, or that you don't really exist.

 

[Jon R]

Hi,
Thank you very much for taking the time to reply.
We’re sorry re the leap day error, you’re quite correct.
But still quite a rare occurrence by what your calculations look like ending up with.
We’re very grateful to all who’ve looked at this for us, and so promptly too.
Many thanks
Jon & Family.

 

[Steven G]

The moment I saw that you had a birthday in January and some after February I too knew that what you said could not be true on leap years. The fact that you said otherwise maybe you don't really exist.

 

[Romsek]

The moment I saw that you had a birthday in January and some after February I too knew that what you said could not be true on leap years. The fact that you said otherwise maybe you don't really exist.