Q.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

For solving the problem we have to calculate (d) first.

d)The probability P(E) that a person voted in the elections, is P(E) = $0.4862$

a) $33.11\%$

b)$38.26\%$

c)$28.63\%$

Step 1: Given Information (part d)

 What percent of voters participated in the local election? 

Step 2: Explanation (part d)

Consider events:

I - a randomly chosen person is an independent

L - a randomly chosen person is a Liberal

C - a randomly chosen person is a  conservative

E - a randomly chosen person participated in the elections.

Given probabilities,

$P\left(I\right) = 0.46$

$P\left(L\right)=0.30$

$P\left(C\right)=0.24$

$P(E/I)=0.35$

$P(E/L)=0.62$

$P(E/C)=0.58$

$\text{a)}P(I\mid E)=\text{? b)}P(L\mid E)=\text{?c)}P(C\mid E)=\text{? d}P\left(E\right)=\text{?}$

Calculate (d) first.

$I,L\text{, and }C\text{ are mutually exclusive (competing hypothesis), therefore: }$

$P\left(E\right)=P(E\mid I)P\left(I\right)+P(E\mid L)P\left(L\right)+P(E\mid C)P\left(C\right)$

$=0.35\cdot 0.46+0.62\cdot 0.30+0.58\cdot 0.24$

$=0.4862$

Step 3: Final Answer (part d)

Percent of voters who participated in the local election is $0.4862$$0.4862$

Step 4: Given Information (part a)

what is the probability that he or she is an Independent?

Step 5: Explanation (part a)

By using the definition of conditional probability, and transforming it to obtain $P\left(IE\right)=P(E/I)P\left(I\right)$

$P(I\mid E)=\frac{P\left(IE\right)}{P\left(E\right)}$

$=\frac{P(E\mid I)\cdot P\left(I\right)}{P\left(E\right)}$

$=\frac{0.35\cdot 0.46}{0.4862}$

$=0.3311$

Step 6: Final Answer (part a)

The probability that he or she is Independent will be $0.3311$

Step 7: Given Information (part b)

what is the probability that he or she a Liberal?  

Step 8: Explanation (part b)

Similarly 

$P(L\mid E)=\frac{P\left(LE\right)}{P\left(E\right)}$

$=\frac{P(E\mid L)\cdot P\left(L\right)}{P\left(E\right)}$

$\frac{0.62.0.30}{0.4862}$

$0.3826$

Step 9: Final Answer (part b)

The probability that he or she a Liberal is $0.3826$ 

Step 10: Given Information (part c)

what is the probability that he or she is a Conservative?

Step 11: Explanation (part c)

Similarly for (c ),

$P(C\mid E)=\frac{P\left(CE\right)}{P\left(E\right)}$

$=\frac{P(E\mid C)\cdot P\left(C\right)}{P\left(E\right)}$

$=\frac{0.58\cdot 0.24}{0.4862}$

$0.2863$

Step 12: Final Answer (part c)

The probability that he or she is a Conservative is $0.2863$