the probability of having more girls than boys in a family of 6 children
$$\\P(4 girls) = 6C4 * 0.5^4*0.5^2 = 15*0.5^6\\ P(5 girls) = 6C5 * 0.5^5*0.5^1 = 6*0.5^6\\ P(6 girls) = 6C6 * 0.5^6*0.5^0 = 1*0.5^6\\\\ p(more\; girls\; than \;boys)=(15+6+1)*0.5^6=\frac{22}{64}=\frac{11}{32}$$
