Solution: the fourth serve, A serves twice and B serves once. A gets 1, and B gets two points as follows:

Serve: A, B, B.

Probability of score: a, B and B: 0.6×0.4×0.6=0. 144.

Score: b, a, b probability: 0.4×0.6×0.6=0. 144.

Scores: B, B and A Probability: 0.4×0.4×0.4=0.064.

Therefore, the probability that the ratio of Party A to Party B is 1: 2 is 2×0. 144+0.064=0.352.