3.18 - A First Course in Probability

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? 

Step-by-Step Solution

Answer

a. $0.3311$ 

b. $0.3825$

c. $0.2863$

d. $24.32\%$

Step 1 : 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

Step 2 : Formula used

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

Step 3 : 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$=\frac{0.161x}{0.4862x}=\frac{805}{2431}=0.3311$

Step 4 : 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$=\frac{0.186x}{0.4862x}=\frac{930}{2431}=0.3825$

Step 5 : 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$=\frac{0.1392x}{0.4862x}=\frac{696}{2431}=0.2863$

Step 6 : Calculation of solution (Part d)

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

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