Suppose that economic outcomes can be classified as either good or bad. Governments differ in ability and this affects the likelihood of good outcomes. There are two types of governments: high ability or low ability. The prior probability that a government is high ability is 1/2. The probability that the economy is good given that the government is high ability is 3/4 while the probability that the economy is good given that the government is low ability is 1/4.
In this case, the incumbent government can manipulate the economy and the electorate will learn (update) their beliefs about the ability of the incumbent government based on the observed state of the economy.
Suppose that the opposition is a high type with probability 1/2. Voters vote for the government with the highest probability of being of a high type.
What is the probability that the incumbent government will win an election against the opposition if the economy is good?