if you have four children

• The total number of permutations of the four children genders is $2^4 = 16$ .
• The total number of combinations of getting two boys and two girls out of four children is $\dbinom 42 = \dfrac {4\times 3}2 = 6$ .
• Therefore the probability of having two boys and two girls out of four children is $\dfrac 6{16} = \boxed{\frac 38}$ .

$P_{boy}=P_{girl}=\frac{1}{2}$

$(b+g)^4=b^4+4b^3g+6b^2g^2+4bg^3+g^4$

The term $6b^2g^2$ represents 2 boys and 2 girls.

$P_{2 boys, 2 girls}=6\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^2=\boxed{\dfrac{3}{8}}$