2023 ACJC CA1A Q2

In a football league, United and City are two teams. They play against each other twice in every season. Their games end in either a win for United or a win for City. The probability that United beats City in their first game of each season is \({0.8}\). If United beats City in their first game, the probability that they will beat City in their second game is \({0.7}\). The probability that these two games in each season have different winners is \({0.4}\).

1. Draw a tree diagram to represent the situation above for one season, and hence show that the probability of United winning their second game given that City won their first game is \({0.8}\).[2]
2. Find the probability that in four seasons (with two games against each other per season), City beats United just twice.[3]