Question

a survey asked people what alternate transportation modes they use. the results are below.
25% of people answered light rail.
12% of people answered walk.
24% of people answered bus.
12% of people answered light rail and bus.
9% of people answered walk and bus.
3% of people answered walk and light rail.
3% of people answered all three.
fill in the following venn diagram with the percents that each region represents then answer the questions below.
what percent of people answered exactly one of the options?
what percent of people did not answer walk, light rail, or bus?
what percent of people answered walk, light rail, or bus?
what percent of people answered walk or bus, but not light rail?
what percent of people answered light rail and bus?
question help: video

Explanation:

Step1: Identify the given percentages

Let \(L\) represent light - rail, \(W\) represent walk, and \(B\) represent bus. We have \(P(L)=25\%\), \(P(W) = 24\%\), \(P(B)=12\%\), \(P(L\cap B)=5\%\), \(P(W\cap B)=9\%\), \(P(W\cap L)=9\%\), and \(P(W\cap L\cap B)=3\%\).

Step2: Calculate the non - overlapping parts of the intersections

For \(W\cap L\) only: \(P((W\cap L)\setminus B)=P(W\cap L)-P(W\cap L\cap B)=9 - 3=6\%\) (Region II).
For \(W\cap B\) only: \(P((W\cap B)\setminus L)=P(W\cap B)-P(W\cap L\cap B)=9 - 3=6\%\) (Region IV).
For \(L\cap B\) only: \(P((L\cap B)\setminus W)=P(L\cap B)-P(W\cap L\cap B)=5 - 3=2\%\) (Region VI).

Step3: Calculate the non - overlapping parts of the single sets

For \(W\) only: \(P(W\setminus(L\cup B))=P(W)-(P((W\cap L)\setminus B)+P((W\cap B)\setminus L)+P(W\cap L\cap B))=24-(6 + 6+3)=9\%\) (Region I).
For \(L\) only: \(P(L\setminus(W\cup B))=P(L)-(P((W\cap L)\setminus B)+P((L\cap B)\setminus W)+P(W\cap L\cap B))=25-(6 + 2+3)=14\%\) (Region III).
For \(B\) only: \(P(B\setminus(W\cup L))=P(B)-(P((W\cap B)\setminus L)+P((L\cap B)\setminus W)+P(W\cap L\cap B))=12-(6 + 2+3)=1\%\) (Region VII).

Step4: Calculate the percentage of people who answered exactly one option

\(P(\text{exactly one})=P(W\setminus(L\cup B))+P(L\setminus(W\cup B))+P(B\setminus(W\cup L))=9 + 14+1=24\%\).

Step5: Calculate the percentage of people who answered walk, light - rail, or bus

\(P(W\cup L\cup B)=P(W\setminus(L\cup B))+P(L\setminus(W\cup B))+P(B\setminus(W\cup L))+P((W\cap L)\setminus B)+P((W\cap B)\setminus L)+P((L\cap B)\setminus W)+P(W\cap L\cap B)=9 + 14+1+6+6+2+3=41\%\).

Step6: Calculate the percentage of people who did not answer walk, light - rail, or bus

\(P((W\cup L\cup B)^C)=100 - P(W\cup L\cup B)=100 - 41 = 59\%\).

Step7: Calculate the percentage of people who answered walk or bus, but not light - rail

\(P(((W\cup B)\setminus L))=P(W\setminus(L\cup B))+P(B\setminus(W\cup L))+P((W\cap B)\setminus L)=9 + 1+6=16\%\).

Step8: Calculate the percentage of people who answered light - rail and bus

\(P(L\cap B)=5\%\).

Answer:

Region I: 9%
Region II: 6%
Region III: 14%
Region IV: 6%
Region V: 3%
Region VI: 2%
Region VII: 1%
Region VIII: 59%
What percent of people answered exactly one of the options? 24%
What percent of people did not answer walk, light - rail, or bus? 59%
What percent of people answered walk, light - rail, or bus? 41%
What percent of people answered walk or bus, but not light - rail? 16%
What percent of people answered light - rail and bus? 5%