Since the probabilities of male and female births are the same, p=½ and the probability of 8 boys out of 10 is the same as 2 girls out of 10.
(p(boy)+p(girl))10=p10+10p9p+45p8p2+...=p10+10p10+45p10+...
We need no more terms than these three which correspond to the probabilities:
All boys, 9 boys + 1 girl, 8 boys + 2 girls, ...
So we simply add the coefficients: 1+10+45=56, giving us 56×½10=56/1024=14/256=0.0547 approx or 5.47%.