One very interesting question and possible approaches
Toss 100 coins, what is the probability of getting more than 60 heads?
The answer to this is (1/2^100)(100C61 + 100C62+...+100C100), where 100Ck denotes 100 choose k.
But how to approximate this number within a minute with only pen and paper?
The first thing that came to my mind is to use Central Limit Theorem which gives probability of 0.05, but for that theorem we need to have that number of tosses tends to infinity, and 100 is far smaller than infinity. Also 0.05 seems too small to me.
Second idea is Integral Moivre-Laplace theorem, but this involves integrating ugly expression (e^(-x2/2)), only not very handy in a few minutes, and number of tosses need to tend to infinity...
So, I would be interested what is the best idea for approaching these kind of approximations?
Any your thoughts would help me a lot.
Thanks!
You can use CLT for n>30. If you don't believe me, try using wolfram alpha for the binomial cdf and compare it to the theoretical normal answer.
EDIT: Also, did you remember to divide by 2 to not include less than 40 heads? I think .05 is too big, actually.
Central Limit Theorem is exactly what you should be using. Since it's 2 SD above 68-95-99.7 would give you 2.5%.
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