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Maths IA help - Birthday Paradox = not enough math?

(Sorry this is so long but really need urgent help! Draft due next week!)


Hi everyone,


So I've been stuck on my Maths IA for a while after having to change topics because my old one was too fiddly. I've discussed doing the Birthday Paradox with my teacher, and she said it was a good topic. However, I'm on holidays now and have sent her an email asking about a problem I was having and possible solutions. In her reply she said she was worried I wasn't doing enough maths.


What I was previously planning on doing in the IA was:


- Discussing the Birthday Paradox itself and the maths behind it (how many people do you need to have in a group so that there is an over 50% probability of 2 people sharing the same birthday - the answer is 23)


- Using the maths of the paradox to calculate how many people you would need for that probability to be 70%, and then 90%


- Investigate whether this works in real life by getting a random sample of 100 of my Facebook friends, and then randomly selecting the number needed for 70% and 90% probabilities and seeing whether it matches up (eg. if the answer for 70% was 30, I would do 10 random selections of 30 of the 100 Facebook friends and see whether 7 out of 10 trials had at least 2 people sharing a birthday)


However, the problem I came across was to do with calculating the number needed for 70% and 90%. Most websites discuss the paradox by using the already known answer for 50% (23) and proving it using a relatively simple calculation. In order to find the number 23 itself, you need to use the Poisson approximation, which my teacher previously said would be too complicated.



I thought that maybe I could get around this either:

a) by selecting a number of people and finding the probability from there (although it would make it more complicated to investigate in terms of real life as it is unlikely it would be a round number like 70%)




b) Using a pre-made graph (which is reliable as the results appear the same on many different websites) of the probabilities from 0% to >99% to estimate the number of people needed for 70% and 90%. I would then use this number (for 70% it looks like its about 30) in the simpler calculation and prove that this number would give a 70% probability.


I know this was really long and thank you for reading it! So basically my question is: would either one of these solutions give me enough Maths to get a good mark on Criterion E: use of mathematics? Or should I try using the Poisson approximation (Maths really isn't a strong subject for me though), or just find another approach altogether? My teacher isn't being particularly detailed in her emails and I have to give the draft in on the first day back at school, so I'd really like an opinion from someone else! Thank you so much in advance


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The paradox is very much about logic. Our teacher showed it us the basics of it in year 9 for fun, I don't think it's mathsy enough or a high enough standard/concept but ask your teacher

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