Question

A total of 46 percent of the voters in a certain city classify themselves as Independents, whereas 30 percent classify themselves as Liberals and 24 percent say that they are Conservatives. In a recent local election, 35 percent of the Independents, 62 percent of the Liberals, and 58 percent of the Conservatives voted. A voter is chosen at random. Given that this person voted in the local election, what is the probability that he or she is

(a) an Independent?

(b) a Liberal?

(c) a Conservative?

(d) What percent of voters participated in the local election?

Short answer

a. $0.3311$

b. $0.3825$

c. $0.2863$

d.$24.32\%$

Step-by-step solution

Given information

The total voters identify as independent is$46\%$and independents voted in the election$35\%$.

The voters identify as liberals$30\%$and liberals voted in the election $62\%$.

The voters identify as conservatives$24\%$and conservatives voted in the election$58\%$.

A voter is chosen at random

Formula used

$\text{Probability of an event}=\frac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}$

Calculation of solution (Part a)

Assume, the total number of people be x.

Number of people who identify as independent =0.46x

Total number of voters$=(0.46\times 0.35x)+(0.3\times 0.62x)+(0.24\times 0.58x)=0.4862x$

Number of independent voters $=0.35\times 0.46x=0.161x$

Probability that a voter selected at random is an independent voter$=\frac{0.161x}{0.4862x}=\frac{805}{2431}=0.3311$

Calculation of solution (Part b)

Number of people who identify as liberal$=0.30x$

Number of liberal voters$=0.62\times 0.30x=0.186x$

Probability that a voter selected at random is a liberal voter$=\frac{0.186x}{0.4862x}=\frac{930}{2431}=0.3825$

Calculation of solution (Part c)

Number of people who identify as conservative$=0.24x$

Number of conservative voters$=0.58\times 0.24x=0.1392x$

Probability that a voter selected at random is a conservative voter$=\frac{0.1392x}{0.4862x}=\frac{696}{2431}=0.2863$

Calculation of solution (Part d)

Total number of voters who participated in the election$=0.2432x$

Percentage of voters who participated in the election$=\frac{0.2432x}{x}\times 100=24.32\%$