My Thanks To …
My thanks go to Elizabeth Miles (web site: http://www.iken.biz/) for prompting this post.
For Your Bright Maths Students
Elizabeth’s question was about how many people’s birthdays can be found in the first 200 million digits of pi. She also thoughtfully provided a link to where you could see if your birthday was in those first 200 million digits: http://www.angio.net/pi/bigpi.cgi.
I realised that this was a problem that I could give to year 12 maths and statistics classes.
A Note About Date Formats
Elizabeth commented that different parts of the world have different formats for dates: mmddyyyy and ddmmyyyy, which raises the question of does the answer change under these circumstances.
A Worked Solution
To save you time (and by way of getting you to check out my working!), I offer the following answer:
- Start with the observation that the distribution of digits appears to be uniform for the purposes of this question: http://mathworld.wolfram.com/PiDigits.html
- There appears to be no correlation between successive digits of pi, nor for any pair of digits at a fixed distance apart.
- Under these conditions, we can infer that any pair, triplet or n-tuple of numbers also occurs uniformly.
- A birthdate, in either ddmmyyyy or mmddyyyy form, is a subset of such octuplets.
- The question then simplifies to identifying how often an arbitrary octuplet occurs, and how many “birthday” octuplets are likely to occur in the first 200 million digits.
- Given any 8 successive digits of pi, the chances of the first pair of digits matching are exactly 0.1, the first and second pairs matching is exactly 0.01, and for all 8 pairs it is 10-8.
- The chances of the octuplet not being in the first 8 digits are 0.99999999
- The chances of the octuplet not being in the first 200 million digits are 0.99999999199999993, or about 0.135
- So about 86% of the population could expect to find their birthdays in the first 200 million digits of pi.
Nice one Phil and thanks for the mention. It’s the sort of thing I would have enjoyed doing with my children when I was a maths teacher (a long time ago).
My sample size, being 5, was small but still 20% not in there.
However it’s worth considering as follows.
Firstly, from this article (see latter part) http://bit.ly/9VoM7A it is not clear whether pi is ‘normal’ in the sense that 0-9 appear equally often (though this could easily be tested for the first 200m digits or any finite subset of digits)
Secondly birthday numbers aren’t random. Days start with 0 1 2 or 3; months go from 01 to 12; years start with 1 or 2.
I was wondering if this might affect the probabilities?
Happy PI Day #pi (via @newscientist)
11:01 AM Mar 14th via web
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Well spotted with my assumptions!
The link in the original article suggests that pi is normal to the first 5×1010 digits.
I think point (6) coupled with points (1-3) above means that working with a subset of all octuplets rather than the whole set would make no difference to the final result.