The registered voters in a certain district break down as follows:

Question

45% Democrats, 42% Republicans, and 13% Independents. In a certain election, 22% of the Democrats, 25% of the Republicans, and 83% of the Independents voted in favor of a proposition. Determine the probability that a randomly chosen voter:

a. Is an Independent.

b. Voted in favor of the proposition.

c. Is and Independent, if the individual voted in favor of the proposition.

d. Not a Democrat, given that the individual did not vote in favor of the proposition.

Answer 1

a.

         Sum of the numbers in question               

Sum of the numbers comprising the whole group

 

Independent voter = 13%               all voters= 45%+42%+13%

 

               13%                         = 13/100=.13=13% chance a randomly chosen voter is Independent.

              100%

b.

22% of the Democrats (45% of the voters) voted for the proposition=9.9% of the voters

25% of the Republicans (42% of the voters) voted for= 10.5% of the voters

83% of the Independents (13%) voted for= 10.79% of the voters= 9.9+10.5+10.79=31.19% of the voters voted for the proposition

31.19%/100%=31.19% chance any one of the voters voted for the proposition.

 

c. 10.79% Independents voted is divided by the percent of voters who voted for the preposition to find the chance a pro-proposition voter is an independent.

 

d. percent of voters who did not vote for the proposition-democrats who did not vote for the proposition divided by the number of voters who did not vote for the proposition.

 

 

Answer 2

a) The probability of a randomly chosen voter being from a specific party is the % of the population that the party makes up. In this case, Independents make up 13% of the population.

 

Answer: 13%

 

for parts b and c, you can either stick with the percentages or you can make up a population number and then calculate the number of people that fall into the given portion (I will use 10000), though the second option is really adding a step for the sake of clarifying what the percentages mean.

 

b) For each party, a certain % voted in favor of the proposition. To reach the total % of the population that voted in favor, you would do the following:

 

% of the population that the party composes * % of that party's voters that voted in favor

Thus for Democrats:

45%*22%=9.9%

OR

4500 (45% of 10000) * 22% = 990/10000 = 9.9%

 

Republicans:

42%*25%=10.5%

OR

4200 (42% of 10000) * 25% = 1050/10000 = 10.5%

 

Independents:

13%*83%=10.79%

OR

1300 (13% of 10000) * 83% = 1079/10000 = 10.79%

 

Add them together:

9.9%+10.5%+10.79%=31.19%

OR

990+1050+1079=3119/10000=31.19%

 

31.19% Voted in favor of the proposition

 

c) The chance of a random voter who voted in favor of the proposition being from a given party would be calculated as follows:

 

% of the population that voted in favor that consists of the party/% of population that voted in favor

For Independents that would be:

10.79%/31.19%=34.59%

OR

1079/3119=34.59%

 

34.59% of the voters who voted in favor are Independents

 

d) If you use the numbers from part b you can cut down on the work you have to do to determine how many people voted against the proposition, by taking the % of population the party makes up and then subtract the % of the population that voted in favor that are from that party, as follows:

 

Democrats

 

45%-9.9%=35.1%

Republican

42%-10.5%=31.5%

Independent

13%-10.79%=2.21%

 

Add them or take the total from part b (31.19%) and subtract it from 100%

either way, you end up with:

68.81%

 

Two ways to do this: Add the % of voters who voted against that are Republican and the % that are Independents, or take the % of voters who voted against that ARE Democrats and subtract that from the total % of voters who voted against

 

First Method:

31.5%+2.21%=33.71%

Second method:

68.81%-35.1%=33.71%

 

Whichever method you use you end up with:

33.71% of the population that voted against the proposition are NOT Democrats

 

The chance of a random voter that voted against and is NOT a Democrat would be determined like this:

% of the population that did NOT vote in favor that are NOT Democrats/% of the population that did NOT vote in favor

 

33.71%/68.81%=48.99%

 

48.99% of the voters who voted against the proposition are NOT Democrats

 

 

I apologize about part d being a bit messy. Comment on my answer if you think the procedure used is not clear enough

Answer 3

a. is given to us, 13%.

 

b. is all the Democrats, Republicans, and Independents who voted in favor of the proposition:

0.45*0.22+0.42*0.25+0.13*0.83=.3119, or 31.19%

 

c. and d. are about conditional probabilities. Let P(A/B) be the probability of event A under the condition B. Then we have the well-known formula

 

P(A/B) = P(A and B)/P(B)

 

c. the condition B is "voted in favor of the proposition", whose probability we found to be .3119

A and B is "independent AND in favor...", whose probability is 0.13*0.83=.1079. Therefore,

 

P=0.1079/0.3119=.346, or 34.6%

 

d. the condition is "did not vote in favor of the proposition", whose probability is 1-0.3119=.6881

A and B is "Not a Democrat AND not in favor", whose probability is 0.42*0.75+0.13*0.17=0.3371. Therefore,

 

P=0.3371/0.6881=0.490, or 49.0%