Two players (player C and player G) are playing a (modified) word guessing game. Both players share the same vocabulary $ V$ and words in $ V$ are grouped into $ K$ bins, denoted as $ b_1$ , $ b_2$ , …, $ b_{K}$ . Furthermore, we know that $ b_{i} \subset V$ , $ \cup_{i=1}^{K} b_i = V$ . Note that those bins may be overlapping and thus there may exist some case where $ b_i \cap b_j \neq \emptyset$ for $ i \neq j$ .
The interaction protocol is described as follows:

Player C uniformly chooses a word $ w$ from the vocabulary $ V$ . Player G does not know which word $ w$ is.

Player G chooses one bin and asks Player C whether his/her chosen word $ w$ is in the bin. If it is, the game ends. Otherwise, Player G will choose another bin.
Questions: What is the best bin choosing order and what is the expected number of times of choosing the bin, according to the best possible order?
Example:
Suppose we have a vocabulary consisting of ten words $ V = \{w_1, w_2, …, w_{10} \}$ and three bins $ b_1 = \{w_1, w_2, …, w_5\}$ , $ b_2 = \{w_6, w_7 \}$ , and $ b_3 = \{w_8, w_9, w_{10} \}$ .
One possible bin choosing order is $ b_1 \rightarrow b_3 \rightarrow b_2$ and the expected number of times of choosing the bin is $ \frac{1}{2}*1 + \frac{1}{2}*\frac{3}{5}*2 + \frac{1}{2}*\frac{2}{5}*\frac{2}{2}*3 = 1.7$ . I suspect this is the best bin choosing order but how can we prove this result?
Notes:
In this problem, we do NOT have the additional knowledge that all bins are nonoverlapping, compared to a related problem.
Thanks.