Question

In Joensuu (Finland), 30 percent of the people are Conservatives, 50 percent are Liberals, and 20 percent are Independents. Records show that in a particular election, 70 percent of the Conservatives voted, 80 percent of the Liberals voted, and 60 percent of the Independents voted.

If a person from Joensuu is selected at random and it is learned that she did not vote in the last election, what is the probability that she is a Liberal?

Use C: person is a Conservative, L: person is a Liberal, I: person is an Independent; and V: person voted.

Answer 1

P[  person is a Conservative ] = P[ C ] = 30% = 0.3

P[ person is a Liberal ] = P[ L ] = 50% = 0.5

P[ person is an Independent ] = P I ] = 20% = 0.2

V: person voted.

P[ V | C ] = 70% = 0.7

P[ V | L ] = 80% = 0.8

P[ V | I ] =60% = 0.6

D be the event person did not voted

P[ D | C ] =1 - P[ V | C ] = 1 - 0.7 = 0.3

P[ D | L ] =1 - P[ V | L ] = 1 - 0.8 = 0.2

P[ D | I ] =1 - P[ V | I ] = 1 - 0.6 = 0.4

P[ D ] = P[ D | C ]*P[ C ] + P[ D | L ]*P[ L ] + P[ D | I ]*P[ I ]

P[ D ] = 0.3*0.3 + 0.2*0.5 + 0.4*0.2

P[ D ] = 0.09 + 0.10 + 0.08

P[ D ] = 0.27

We need to find  she did not vote in the last election, what is the probability that she is a Liberal?

P[ L | D ] = P[ D | L ]*P[ L ] / P[ D ]

P[ L | D ] = 0.2*0.5 / 0.27

P[ L | D ] = 0.1/0.27

P[ L | D ] = 0.37