Question

In a certain city, 30 percent of the people are Conservatives,50 percent are Liberals, and 20 percent are Independents. Records show that in a particular election, 65percent of the Conservatives voted, 82 percent of the Liberals voted, and 50 percent of the Independents voted. If a person in the city 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?

Short answer

If a person in the city is selected at random and it is learned that she did not vote in the last election, the probability that she is Liberal is 0.30.

Step-by-step solution

Given information

Here said that there are 30% conservatives, 50% Liberals, and 20% Independents people of the whole population in a certain city.

By a record, we know that 65% of conservatives, 82% of Liberals, and 50% of Independents are voted in a particular vote.

State the events

Let us consider three events,\(C,L,\;and\;I\)for selecting a Conservative, Liberal, and Independent, person respectively. And also consider the event\(V\)for voting.

So,\(P\left( C \right) = 0.3\),\(P\left( L \right) = 0.5\),\(P\left( I \right) = 0.2\)

Now we have the probability that a person voted and she is Conservative,\(P\left( {V|C} \right) = 0.65\), and for Liberal it is\(P\left( {V|L} \right) = 0.82\)and for Independent it is\(P\left( {V|I} \right) = 0.50\)

So, the probability of a person did not vote but given that she is Conservative is \(\)\(\begin{aligned}{}P\left( {{V^c}|C} \right) &= 1 - 0.65\\ &= 0.35\end{aligned}\)

And similarly, we get, \(P\left( {{V^c}|L} \right) = 0.18\) and \(P\left( {{V^c}|I} \right) = 0.50\)

Calculate the probability

Now the probability that a randomly selected person is Liberal, given that she is not voted.

\(\begin{aligned}{}{\bf{Pr}}\left( {{\bf{L|}}{{\bf{V}}^{\bf{c}}}} \right){\bf{ = }}\frac{{{\bf{Pr}}\left( {\bf{L}} \right){\bf{Pr}}\left( {{{\bf{V}}^{\bf{c}}}{\bf{|L}}} \right)}}{{{\bf{Pr}}\left( {\bf{C}} \right){\bf{Pr}}\left( {{{\bf{V}}^{\bf{c}}}{\bf{|C}}} \right){\bf{ + Pr}}\left( {\bf{L}} \right){\bf{Pr}}\left( {{{\bf{V}}^{\bf{c}}}{\bf{|L}}} \right){\bf{ + Pr}}\left( {\bf{I}} \right){\bf{Pr}}\left( {{{\bf{V}}^{\bf{c}}}{\bf{|I}}} \right)}}\\ = \frac{{0.18 \times 0.5}}{{\left( {0.3 \times 0.35} \right) + \left( {0.5 \times 0.18} \right) + \left( {0.2 \times 0.50} \right)}}\\ = 0.30\end{aligned}\)

So, there are 30% chance of that the randomly selected person who is noted voted, is Liberal.