# Bayes Theorem and probability

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?

• I cam up with an answer 1/4. But this seems too easy so I am assuming it is wrong. Given that the probability of the opposition being High type is 1/2 and the probability of the economy being good when given that we know the opposition is high ability tells me that the answer to this is 3/4 or 75%. So the chances of the incumbent government winning against the opposition will surely be 1/4? If i am wrong, which is quite likely please explain it to me as to how i would go about working out. – Dimoftie Cristina Diana Feb 14 '14 at 12:58
• I'd say, if good governments are associated with good economies more than with bad ones (75% to 25%), then surely if the economy is good, voters will conclude it's more than likely due to good government and will be more likely to vote for the government than the opposition. – TooTone Feb 14 '14 at 13:30
• The other thing is that the question says voters vote for whichever party has the highest probability of being a high type. This seems odd to me. If the probability of the government being high type (given the good economy) is greater than the opposition being high type, then voters will vote for govt with probability 1. In my answer I haven't addressed this, I've just looked at the probability of high ability given good economy. – TooTone Feb 14 '14 at 13:31

Let $G$ be the event that government is high ability. We are given $\mathbb{P}(G)=1/2$.

Let $E$ be the event that economy is good. We are given $\mathbb{P}(E|G)=3/4$ and $\mathbb{P}(E|\neg G)=1/4$.

Bayes theorem is

$$\mathbb{P}(B|A) = \frac{\mathbb{P}(A|B) \mathbb{P}(B)}{ \mathbb{P}(A)}$$

and often the theorem of total probability proves useful on the denominator

$$\mathbb{P}(A) =\mathbb{P}(A|B)\mathbb{P}(B) + \mathbb{P}(A|\neg B)\mathbb{P}(\neg B)$$

Can you see how to plug in the probabilities given in the question and get the probability of [the electorate believing that] the government is high ability given the economy is good?

• The original is worded like a homework problem, and shows no work by the OP, which is probably the reason for down-votes there. You have provided a hint, great. Now: can other over-eager answerers restrain themselves until the OP gives it a try???? – GEdgar Feb 14 '14 at 14:19
• HI @GEdgar, thanks for your comment. I'm still relatively new to the site and would appreciate some feedback. Do you think I went too far with my answer or was your concern that I / others definitely went no further than this answer? (I did a search following your comment and found this.) – TooTone Feb 14 '14 at 15:43
• Your reply was good. From experience, I fear that someone else will go further and provide a compete answer, not waiting for Dimoftie to come back and read your answer. – GEdgar Feb 14 '14 at 21:19
• @GEdgar thanks, good to know either way – TooTone Feb 14 '14 at 21:20