Question

question 7 (multiple choice worth 5 points) (02.09h mc) a survey from the department of motor vehicles was given to residents in the city of smallville. the results showed that 80% held a valid driver’s license. of the individuals who held a valid driver’s license, 20% had at least one traffic violation, whereas 30% who did not have a valid driver’s license had at least one traffic violation. what percent of smallville residents have at least one traffic violation? 50% 46% 36% 22%

Explanation:

Step1: Define variables

Let total residents be \( 100 \) (for percentage calculation). Let \( L \) be those with license, \(
eg L \) without. \( V \) be those with violation, \(
eg V \) without. Given \( P(L) = 80\% \), so \( P(
eg L)=20\% \). \( P(V|L) = 20\% \), \( P(V|
eg L)=30\% \).

Step2: Apply law of total probability

The formula for total probability of \( V \) is \( P(V)=P(V|L)P(L)+P(V|
eg L)P(
eg L) \). Substitute values: \( P(V)=(0.20\times0.80)+(0.30\times0.20) \).

Step3: Calculate each term

First term: \( 0.20\times0.80 = 0.16 \). Second term: \( 0.30\times0.20 = 0.06 \).

Step4: Sum the terms

Add the two results: \( 0.16 + 0.06 = 0.22 \), which is \( 22\% \). Wait, no—wait, the question is "What percent of Smallville residents have at least one traffic violation?" Wait, no, wait the question in the image: "Of the individuals who held a valid driver’s license, 20% had at least one traffic violation, whereas 30% who did not have a valid driver’s license had at least one traffic violation. What percent of Smallville residents have at least one traffic violation?" Wait, no, wait the initial problem: Wait, no, maybe I misread. Wait, the user's problem: Wait, the question is to find the percentage of residents with at least one traffic violation. Wait, let's re-express:

Total residents: 100.

With license: 80 (80% of 100). Of these, 20% have violation: \( 0.20\times80 = 16 \).

Without license: 20 (20% of 100). Of these, 30% have violation: \( 0.30\times20 = 6 \).

Total with violation: \( 16 + 6 = 22 \). So 22%? But wait, the options include 22% as one of them (the last option? Wait the options are 50%, 46%, 36%, 22%? Wait the image shows options: 50%, 46%, 36%, 22% (from left to right? Wait no, the first option is 50%, then 46%, 36%, 22%? Wait no, the user's image: the first option is 50%, second 46%, third 36%, fourth 22%? Wait no, the text in the image: "50%", "46%", "36%", "22%". Wait, but my calculation gave 22%, but wait, maybe I misread the question. Wait, the question is: "What percent of Smallville residents have at least one traffic violation?" Wait, no, wait the problem says: "80% held a valid driver’s license. Of the individuals who held a valid driver’s license, 20% had at least one traffic violation, whereas 30% who did not have a valid driver’s license had at least one traffic violation. What percent of Smallville residents have at least one traffic violation?" Wait, no, maybe the question is different? Wait, no, the calculation: 80% with license, 20% of them have violation: 0.80.2=0.16. 20% without license, 30% of them have violation: 0.20.3=0.06. Total: 0.16+0.06=0.22, so 22%. So the answer is 22%, which is the last option (22%).

Wait, but let me check again. Let total population be 100.

• License holders: 80. Violation in license holders: 20% of 80 = 16.

• Non-license holders: 20. Violation in non-license holders: 30% of 20 = 6.

Total violation: 16 + 6 = 22. So 22% of residents have violation. So the answer is 22%, which is the option with 22%.

Answer:

22% (the option with 22%)