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[ people are Conservatives ] = P[ C ] =30% = 0.3

P[ people are Liberals ] = P[ L ] =50% = 0.5

P[ people are Independents ] = P[ I ] =20% = 0.2

V: person voted

P[ V | L ] = 80% = 0.8

P[ V | C ] = 70% = 0.7

P[ V | I ] = 60% = 0.6

D: did not vote

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

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

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

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

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

P[ D ] = 0.10 + 0.09 + 0.08

P[ D ] = 0.27

We need to find

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

P[ L | D ] = 0.10 / 0.27

P[ L | D ] = 0.37