Presentation Problem 3.7

3.7 Swing voters. A Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. 35% of respondents identified as Independent, 23% identified as swing voters, and 11% identified as both.
(a) Are being Independent and being a swing voter disjoint, i.e. mutually exclusive?
(b) Draw a Venn diagram summarizing the variables and their associated probabilities.
(c) What percent of voters are Independent but not swing voters?
(d) What percent of voters are Independent or swing voters?
(e) What percent of voters are neither Independent nor swing voters?
(f) Is the event that someone is a swing voter independent of the event that someone is a political Independent?

Solution 3.7

3.7(a)

Being Independent and Swing Voters are not disjoint (i.e not mutually exclusive). From the information provided, 11% identified as both Independent and swing voters which means that they can both occur at the same time. Hence, they are mutually non-exclusive.

3.7(b)

Draw a Venn Diagram

Sample size = 2373
Independent (I) = 35% = 0.35
Swing Voters (S) = 23% = 0.23
Both Independent and Swing Voters (I n S) = 11% = 0.11

Venn diagram

library(VennDiagram)
independent <- 0.35
swing <- 0.23
both <- 0.11
grid.newpage()
Venn <- draw.pairwise.venn(area1 = independent, area2 = swing, cross.area = both, category = c("Independent", "Swing"))

3.7(c)

Percent of Americans that are Independent but not swing voters

P(I) = Probability of Independent voters = 0.35
P(S) = Probability of Swing voters = 0.23
P(Sc) = Probability of non-swing voters = 1 - P(S) = 1 - 0.23 = 0.77
P(Independent but not swing voters) = P(I and Sc)
= P(I) x P(Sc) = 0.35 x 0.77 = 0.2695
This means that about 26.95% are Independent but not swing voters

3.7(d)

Percent of voters are Independent (I) or Swing (S) voters
P(I or S) = P(I) + P(S) - P(I n S) = 0.35 + 0.23 - 0.11 = 0.47

47% of the voters are Independent or Swing voters

3.7(e)

Percent of voters who are neither Independent nor swing voters P(I or S)c = 1 - P(I or S) = 1 - 0.47 = 0.53
This means that about 53% are neither Independent nor swing voters.

3.7(f)

Yes it is independent because just knowing that someone is Independent does not provide additional information about whether they are swing or non-swing voters.