Question

You are counting ballots at an election. However, you lost a sheet of paper and now you are not quite sure which district one of the ballot boxes came from. You know it is either district A or district B and you think there is about 50% chance that it is from either district. District A leans republican and typically 60% of the voters in this district vote republican and 40% vote democratic. District B leans democratic and typically 60% of the voters in the this district vote for the democratic party and the rest vote republican. Now you count the ballots and find that there are 32 democratic votes and 48 republican votes. What is the probability that the ballot box came from district A?

Answer 1

Answer

Here let us assume A, B , D and R are four events such that, for any particular random ballot box,

A=Event that it comes from district A

B=Event that it comes from district B

D=Event that it leans to democratic

R=Event that it leans to republican.

Now by the problem, there is about 50% chance that it is from either district. District A leans republican and typically 60% of the voters in this district vote republican and 40% vote democratic. District B leans democratic and typically 60% of the voters in the this district vote for the democratic party and the rest vote republican.

So that means, P(A)=P(B)=0.5

P(D|A)=0.4 , P(R|A)=0.6 , P(D|B)=0.6 , P(R|B)=0.4

Now we count the ballots and find that there are 32 democratic votes and 48 republican votes. That is the ballot box leans to republican (and with 60% proportion but we cannot say for sure that it comes from A)

So the probability that the ballot box came from district A

=P(A|R)

$=\frac{P(A\cap R)}{P(R)}=\frac{P(R|A).P(A)}{P(R|A).P(A)+P(R|B).P(B)}$ (Bayes' theorem)

$=\frac{0.6\times 0.5}{0.6\times 0.5+0.4\times 0.5}=0.6$

SO the probability is 60%

Hence the answer.............

Thank you...........