2017 NS 24

\(P[\geq 2\text{ heads in either compartment}]= \\ 2P[\geq 2\text{ heads in a given compartment}]=\\ 2\left(P[\geq 2\text{ heads|3 coins in compartment}]P[\text{3 coins in compartment}] +\\ P[\geq 2\text{ heads|2 coins in compartment}]P[\text{2 coins in compartment}] \right)=\\ 2\left(\left(\left(\dfrac 1 8 + \dbinom{3}{2}\dfrac 1 8\right)\dfrac 1 8\right)+\dfrac 1 4 \dbinom{3}{2}\dfrac 1 8\right) = \\ 2\left(\dfrac{1}{16}+\dfrac{3}{32}\right)= \dfrac{5}{16}\)