Question

a survey asked people what alternative transportation modes they use. the results are below. 23% of people answered carpool. 24% of people answered bus. 27% of people answered walk. 10% of people answered bus and carpool. 13% of people answered walk and bus. 9% of people answered walk and carpool. 5% 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 answered bus and carpool? % what percent of people answered walk or carpool, but not bus? % what percent of people answered walk, bus, or carpool? % what percent of people did not answer walk, bus, or carpool? % question help: video

Explanation:

Step1: Recall the principle of Venn - diagram intersection

Let \(W\) be the set of people who walk, \(B\) be the set of people who take the bus, and \(C\) be the set of people who carpool. We know that \(n(W) = 10\%\), \(n(B)=27\%\), \(n(C) = 23\%\), \(n(B\cap C)=13\%\), \(n(W\cap B)=19\%\), \(n(W\cap C)=5\%\), and \(n(W\cap B\cap C)=5\%\).

Step2: Calculate the non - overlapping parts of the intersections

For the intersection of two sets, we use the formula \(n(A\cap B)=n(A\cap B\cap\overline{C})+n(A\cap B\cap C)\).

• \(n(B\cap C\cap\overline{W})=n(B\cap C)-n(W\cap B\cap C)=13 - 5=8\%\)
• \(n(W\cap B\cap\overline{C})=n(W\cap B)-n(W\cap B\cap C)=19 - 5 = 14\%\)
• \(n(W\cap C\cap\overline{B})=n(W\cap C)-n(W\cap B\cap C)=5 - 5=0\%\)

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

• \(n(W\cap\overline{B}\cap\overline{C})=n(W)-n(W\cap B\cap\overline{C})-n(W\cap C\cap\overline{B})-n(W\cap B\cap C)=10-(14 + 0+5)= - 9\) (This is wrong. Let's start over using the correct formula \(n(A)=n(A\cap\overline{B}\cap\overline{C})+n(A\cap B\cap\overline{C})+n(A\cap\overline{B}\cap C)+n(A\cap B\cap C)\))
• \(n(W\cap\overline{B}\cap\overline{C})=n(W)-(n(W\cap B)+n(W\cap C)) + n(W\cap B\cap C)=10-(19 + 5)+5=-9\) (wrong). Correctly: \(n(W\cap\overline{B}\cap\overline{C})=n(W)-(n(W\cap B)+n(W\cap C)-n(W\cap B\cap C))=10-(19 + 5 - 5)=1\%\)
• \(n(B\cap\overline{W}\cap\overline{C})=n(B)-(n(B\cap W)+n(B\cap C)-n(W\cap B\cap C))=27-(19 + 13 - 5)=0\%\)
• \(n(C\cap\overline{W}\cap\overline{B})=n(C)-(n(C\cap W)+n(C\cap B)-n(W\cap B\cap C))=23-(5 + 13 - 5)=10\%\)

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

The percentage of people who answered exactly one option is \(n(W\cap\overline{B}\cap\overline{C})+n(B\cap\overline{W}\cap\overline{C})+n(C\cap\overline{W}\cap\overline{B})=1 + 0+10 = 11\%\)

Step5: Calculate the percentage of people who answered bus and carpool

We are given \(n(B\cap C)=13\%\)

Step6: Calculate the percentage of people who answered walk or carpool, but not bus

\(n((W\cup C)\cap\overline{B})=n(W\cap\overline{B}\cap\overline{C})+n(C\cap\overline{W}\cap\overline{B})=1+10 = 11\%\)

Step7: Calculate the percentage of people who answered walk, bus, or carpool

We use the principle of inclusion - exclusion: \(n(W\cup B\cup C)=n(W)+n(B)+n(C)-n(W\cap B)-n(W\cap C)-n(B\cap C)+n(W\cap B\cap C)=10 + 27+23-19 - 5 - 13+5=28\%\)

Step8: Calculate the percentage of people who did not answer walk, bus, or carpool

\(100 - n(W\cup B\cup C)=100 - 28 = 72\%\)

The Venn - diagram regions:

• \(I=n(W\cap\overline{B}\cap\overline{C}) = 1\%\)
• \(II=n(W\cap B\cap\overline{C})=14\%\)
• \(III=n(B\cap\overline{W}\cap\overline{C})=0\%\)
• \(IV=n(W\cap C\cap\overline{B})=0\%\)
• \(V=n(W\cap B\cap C)=5\%\)
• \(VI=n(B\cap C\cap\overline{W})=8\%\)
• \(VII=n(C\cap\overline{W}\cap\overline{B})=10\%\)
• \(VIII=72\%\)

Answer:

I. 1%
II. 14%
III. 0%
IV. 0%
V. 5%
VI. 8%
VII. 10%
VIII. 72%
What percent of people answered exactly one of the options? 11%
What percent of people answered bus and carpool? 13%
What percent of people answered walk or carpool, but not bus? 11%
What percent of people answered walk, bus, or carpool? 28%
What percent of people did not answer walk, bus, or carpool? 72%