The probability that one taxpayer filed is 4/5 and the probability that the taxpayer didn't file is 1/5. Obviously a taxpayer must do one or the other and the probability of either/or is a certainty=1. Using the binomial expansion of (a+b)^n, we can put a=4/5 and b=1/5 and n=15. The probability that a taxpayer either filed or didn't file is (a+b), which is always 1. The binomial expansion adds together as a series all the various combinations of probabilities and the general term, r, is given by (n(n-1)(n-2)...(n-r+1)a^(n-r+1)b^(r-1))/(1*2*3*...*r), which can be written (nCr)a^(n-r+1)b^(r-1), using the symbol for combination. 15C7 and 15C8 have the same value of 6435. So the probability of 7 or 8 filing is 6435(0.8^8*0.2^7 + 0.8^7*0.2^8)=6435*(0.8*0.2)^7=0.0173 approx or 1.73%.
