Question

Step by Step Bayes Theorem In an imagined survey of voters, you found 55 % of the voters were Democrats and the rest were Republicans. You also found 90 % of the Democrats voted President Obama but only 14 % of the Republicans did. 55%, or, P(D) 0.55. First some easy questions. You clearly know that the probability voter is a Democrat What is the probability a voter is a Republican, P(D')? What is the probability of somebody voting for Obama given the voter is a Democrat, P(OD)? What is the probability somebody voting for Obama given the voter is a Republican, P(OID'? Now a harder question (use the law of total probability). What is the probability a random voter voted for Obama, P(O)? Finally, the Bayes Theorem Question. In a follow-up interview with a randomly picked voter, you find out the voter voted for Obama. What is the probability that voter is a Democrat given that you know they voted for Obama, P(DIO)? A interesting note: your evaluation of the probability that a random person is a Democrat, P(D), may change if you are given more information about that person, P(DIO).

Answer 1

Solution: Given, P(D) 0.55 90% of the Democrats voted for President Obama P(0 D) 0.90 14% of the Republicans voted for President Obama. P(0|D') 0.14 - Probability that the voter is a Republican: P(D') 1 P(D) 1 0.55 0.45 Probability of somebody voting for Obama given that the voter is a Democrat: P(0 D) 0.90 Probability of somebody voting for Obama given that the voter is a Republican: P(0|D') 0.14 Probability that a random voter voted for Obama: P(0) P(On D) P(0 n D') P(0) P(0|D) x P(D) P(0|D') x P(D') P(0) 0.90 x 0.55 0.14 x 0.45 P(0) 0.5580 Probability that the voter is a Democrat given that the voter voted for Obama = P(D|0) From Baye's theorem: P(D) x P(o D) P(D) x P(o D) P(D') x P(0|D') P(DIO) 0.55 x 0.90 P(DI0) 0.55 x 0.90 0.45 x 0.14 P(DIO) 0.8871