Determining whether a coin is fair I have a dataset where an ostensibly 50% process has been tested 118 times and has come up positive 84 times.
My actual question:


*

*IF a process has a 50% chance of testing positive and

*IF you then run it 118 times

*What is the probability that you get AT LEAST 84 successes?


My gut feeling is that, the more tests are run, the closer to a 50% success rate I should get and so something might be wrong with the process (That is, it might not truly be 50%) but at the same time, it looks like it's running correctly, so I want to know what the chances are that it's actually correct and I've just had a long string of successes.

Of course, 118 is in the "small numbers regime", where one

can easily (use a computer to) calculate the probability exactly.

By wolframalapha,

the probability that you get at least 84 successes $\;\;=\;\; \frac{\displaystyle\sum_{s=84}^{118}\:\binom{118}s}{2^{118}}$
$=\;\; \frac{392493659183064677180203372911}{166153499473114484112975882535043072} \;\;\approx\;\; 0.00000236224 \;\;\;\; $.
