Expected Questions in a Ternary-Then-Binary Strategy

A strategy first asks a ternary question that isolates state A on one branch, state B on another branch, and groups states C and D on the last branch. If the prior over [A, B, C, D] is [0.4, 0.3, 0.2, 0.1], what is the expected number of questions if the grouped branch uses one extra yes/no question?