Venn diagrams and counting

Learning goals

Represents overlapping events in Venn diagrams.
Places counts in mutually exclusive regions correctly.
Uses totals to determine missing region counts.
Computes union, intersection, and complement counts from the diagram.

Difficulty: 1 – 5

Click problems to add them to your worksheet.

Venn diagrams and counting 1

In the Venn diagram, the letters \(P\), \(Q\), \(R\), and \(S\) mark the four regions. Match them with “only \(A\),” “\(A\cap B\),” “only \(B\),” and “outside \(A\cup B\).”

Hints

- Describe the location of each letter relative to both circles. - Begin with the region that is clearly outside both circles. - Use the rectangle as the boundary of the full sample space.

Solution

1. Point \(P\) lies inside \(A\) but outside \(B\), so it represents only \(A\). 2. Point \(Q\) lies in the overlap, so it represents \(A\cap B\). 3. Point \(R\) lies inside \(B\) but outside \(A\), so it represents only \(B\). 4. Point \(S\) lies inside the universal-set rectangle but outside both circles, so it represents the region outside \(A\cup B\).

Answer

\(P\): only \(A\); \(Q\): \(A\cap B\); \(R\): only \(B\); \(S\): outside \(A\cup B\).

Venn diagrams and counting 2

The rectangle represents the sample space \(S\). The shaded part of the Venn diagram is the event being described. Write the shaded event using \(A\) and \(B\).

Hints

- Does the shading include the overlap of the two circles? - Decide whether the diagram represents “and” or “or.”

Solution

1. Every point inside \(A\) is shaded, including the overlap. 2. Every point inside \(B\) is also shaded. 3. Therefore, the shaded event consists of outcomes in \(A\) or \(B\), which is \(A\cup B\).

Answer

\(A\cup B\).

Venn diagrams and counting 3

Point X marks one outcome. Which event contains X: \(A\cap B\), \(A\cap B^c\), or \(A^c\cap B\)?

Hints

- Decide whether X is inside A, B, or both. - Which set operation means an outcome belongs to both events?

Solution

1. Point X lies inside both circles A and B. 2. The outcomes common to both events form \(A\cap B\).

Answer

\(A\cap B\)

Venn diagrams and counting 4

A school surveyed \(255\) students about taking French and Italian. The totals next to the circles count everyone taking each language; the number in the overlap counts students taking both. How many students take only French, only Italian, and neither language?

Hints

- Which number lies inside both circles? - Each circle's total includes the overlap. - How can you use the full survey total to find the region outside both circles?

Solution

1. The diagram gives \(94\) French students, \(66\) Italian students, and \(25\) in both groups. 2. Only French: \(94-25=69\). Only Italian: \(66-25=41\). 3. All four regions total \(255\). Neither: \(255-(69+25+41)=120\).

Answer

Only French: \(69\). Only Italian: \(41\). Neither: \(120\).

Venn diagrams and counting 5

Four labeled points represent possible outcomes. List the points in each event. a) \(A^c\cap B\) b) \(A\cup B^c\) c) \((A\cup B)^c\)

Hints

- For each point, decide whether it belongs to \(A\), \(B\), both, or neither. - How does the complement in an expression change which points qualify? - Check the point outside both circles as carefully as the points inside.

Solution

1. \(R\) is inside \(B\) but outside \(A\), so part a) contains only \(R\). 2. \(P\) and \(Q\) are inside \(A\), and \(T\) is outside \(B\), so part b) contains \(P,Q,T\). 3. Only \(T\) lies outside both circles but inside the sample space, so part c) contains \(T\).

Answer

a) \(\{R\}\) b) \(\{P,Q,T\}\) c) \(\{T\}\)

Venn diagrams and counting 6

In the Venn diagram, set \(A\) represents students in the school choir and set \(B\) represents students in the school band. The numbers show how many students belong to each region. Explain what each number means and find the total number of students.

Hints

- Look at whether each printed number lies in the left circle, right circle, their overlap, or outside both. - Students in both activities belong to the overlap rather than two separate regions. - Every student should be counted exactly once in the total.

Solution

1. The number \(11\) lies only inside \(A\), so \(11\) students are in choir only. 2. The number \(4\) lies in the overlap, so \(4\) are in both groups. 3. The number \(7\) lies only inside \(B\), so \(7\) are in band only. 4. The number \(3\) lies outside both circles, so \(3\) are in neither group. 5. The total is \(11+4+7+3=25\) students.

Answer

Choir only: \(11\); both groups: \(4\); band only: \(7\); neither: \(3\). Total: \(25\) students.

Venn diagrams and counting 7

The Venn diagram gives the number of elements in each nonoverlapping region. Find \(n(A)\), \(n(B)\), \(n(A\cup B)\), and \(n(S)\).

Hints

- Which parts of the diagram belong to both circles? - The union includes each inside-circle region once. - The full sample space also includes elements outside both circles.

Solution

1. Circle \(A\) contains the exclusive region and the overlap: \(n(A)=18+6=24\). 2. Circle \(B\) contains \(9+6=15\) elements. 3. The union covers all three inside regions, so \(n(A\cup B)=18+6+9=33\). 4. Including the outside region gives \(n(S)=18+6+9+5=38\).

Answer

\(n(A)=24\), \(n(B)=15\), \(n(A\cup B)=33\), and \(n(S)=38\).

Venn diagrams and counting 8

In a Venn diagram, \(n(A)=26\), \(n(B)=19\), \(n(A\cap B)=7\), and \(4\) elements lie outside both circles. Find the number of elements in each of the four regions.

Hints

- Subtract the overlap from each circle total. - Assign each given or calculated value to exactly one region.

Solution

1. The number only in \(A\) is \(26-7=19\). 2. The number only in \(B\) is \(19-7=12\). 3. The overlap contains \(7\) elements. 4. The region outside both circles contains \(4\) elements.

Answer

Only \(A\): \(19\); both: \(7\); only \(B\): \(12\); outside both: \(4\).

Venn diagrams and counting 9

Tarek records students' transportation to school in a Venn diagram. Event \(A\) is “uses a bicycle,” and event \(B\) is “uses the bus.” Tarek writes \(n(A)=14\). Explain and correct his error using the diagram.

Hints

- Circle \(A\) includes more than just the part outside \(B\). - Which other region is still inside \(A\)? - Sum each nonoverlapping region belonging to \(A\) once.

Solution

1. The number \(14\) is in the bicycle-only region, not the entire bicycle circle. 2. The shared region, containing \(6\) students, also belongs to circle \(A\). 3. The correct bicycle total is \(n(A)=14+6=20\).

Answer

Tarek omitted the \(6\) students in the overlap. The correct value is \(n(A)=20\).

Venn diagrams and counting 10

The four regions of a Venn diagram contain the following people: only \(A\): \(\{\text{Lea},\text{Noah}\}\); both sets: \(\{\text{Mina}\}\); only \(B\): \(\{\text{Emil},\text{Sara}\}\); outside both: \(\{\text{Ali}\}\). List the sets \(A\), \(B\), \(A\cup B\), and \(A\cap B\).

Hints

- Read each circle as the combination of its exclusive region and the overlap. - The person outside both circles belongs to neither set.

Solution

1. Set \(A\) contains the “only \(A\)” region and the overlap: \(A=\{\text{Lea},\text{Noah},\text{Mina}\}\). 2. Set \(B\) contains the “only \(B\)” region and the overlap: \(B=\{\text{Mina},\text{Emil},\text{Sara}\}\). 3. The union contains everyone inside at least one circle: \(A\cup B=\{\text{Lea},\text{Noah},\text{Mina},\text{Emil},\text{Sara}\}\). 4. The intersection contains only Mina: \(A\cap B=\{\text{Mina}\}\).

Answer

\(A=\{\text{Lea},\text{Noah},\text{Mina}\}\), \(B=\{\text{Mina},\text{Emil},\text{Sara}\}\), \(A\cup B=\{\text{Lea},\text{Noah},\text{Mina},\text{Emil},\text{Sara}\}\), and \(A\cap B=\{\text{Mina}\}\).

Venn diagrams and counting 11

The four regions of a Venn diagram have probabilities \(0.31\) only in \(A\), \(0.18\) in both sets, \(0.27\) only in \(B\), and \(0.24\) outside both sets. Find \(P(A)\), \(P(B)\), and \(P(A\cup B)\).

Hints

- Include the overlap in each individual event. - For the union, add every region inside at least one circle.

Solution

1. Event \(A\) includes the “only \(A\)” region and the overlap: \(P(A)=0.31+0.18=0.49\). 2. Event \(B\) includes the “only \(B\)” region and the overlap: \(P(B)=0.27+0.18=0.45\). 3. The union includes all three regions inside at least one circle: \(P(A\cup B)=0.31+0.18+0.27=0.76\).

Answer

\(P(A)=0.49\), \(P(B)=0.45\), and \(P(A\cup B)=0.76\).

Venn diagrams and counting 12

In a study group, \(15\) people like puzzles, represented by \(A\); \(12\) people like logic games, represented by \(B\); \(5\) like both; and \(2\) like neither. What four numbers belong in the regions of a Venn diagram?

Hints

- Each circle total already includes the shared group. - Subtract the shared group to find each exclusive region.

Solution

1. The number who like only puzzles is \(15-5=10\). 2. The overlap contains \(5\) people. 3. The number who like only logic games is \(12-5=7\). 4. The outside region contains \(2\) people.

Answer

Only \(A\): \(10\); both: \(5\); only \(B\): \(7\); neither: \(2\).

Venn diagrams and counting 13

The Venn diagram should show \(30\) outcomes in all. There are \(17\) outcomes in \(A\) and \(18\) in \(B\). Exactly one region count in the diagram is wrong. Which one? Find the correct count and explain.

Hints

- Which regions contribute to the total for each circle? - Compare the circle totals and the overall total with the diagram. - Can you change one count while keeping both given circle totals correct?

Solution

1. The circle totals check: \(9+8=17\) in \(A\) and \(8+10=18\) in \(B\). 2. The diagram totals \(9+8+10+5=32\), but the sample space has \(30\) outcomes. 3. The only region outside both circles must therefore contain \(30-(9+8+10)=3\) outcomes. The displayed \(5\) is incorrect.

Answer

The count outside both circles should be \(3\), not \(5\). The circle totals already agree with \(17\) and \(18\), and all four regions must total \(30\).

Venn diagrams and counting 14

In a Venn diagram, \(n(A\cup B)=41\), the “only \(A\)” region contains \(17\) elements, and the “only \(B\)” region contains \(14\) elements. Find the number of elements in the overlap.

Hints

- Separate the union into its three nonoverlapping regions. - Find the missing part of the total inside the circles.

Solution

1. The union consists of three nonoverlapping regions: only \(A\), both sets, and only \(B\). 2. Subtract the two exclusive regions from the union total: \(41-17-14=10\). 3. Therefore, \(n(A\cap B)=10\).

Answer

The overlap contains \(10\) elements.

Venn diagrams and counting 15

A Venn diagram shows \(23\%\) only in \(A\), \(17\%\) in both sets, and \(28\%\) only in \(B\). What percentage lies outside both sets?

Hints

- First add the three regions inside the circles. - Then find the remaining part of the full \(100\%\).

Solution

1. The percentage inside at least one circle is \(23\%+17\%+28\%=68\%\). 2. Subtract from the whole: \(100\%-68\%=32\%\).

Answer

\(32\%\) lies outside both sets.

Venn diagrams and counting 16

The proportions \(0.22\), \(0.31\), \(0.29\), and \(0.18\) are assigned to the Venn-diagram regions “only \(A\),” “both,” “only \(B\),” and “outside both,” respectively. Determine whether this assignment is possible. Then find \(P(A\cap B)\) and \(P(A^c\cap B^c)\).

Hints

- First check whether the four regions make up the entire sample space. - Then match each event expression to its diagram region.

Solution

1. The four nonoverlapping proportions sum to \(0.22+0.31+0.29+0.18=1\), so the assignment is possible. 2. The intersection is the overlap, so \(P(A\cap B)=0.31\). 3. The region outside both circles represents \(A^c\cap B^c\), so \(P(A^c\cap B^c)=0.18\).

Answer

The assignment is possible. \(P(A\cap B)=0.31\), and \(P(A^c\cap B^c)=0.18\).

Venn diagrams and counting 17

In a Venn diagram, \(12\) elements lie only in \(A\), \(7\) lie in both sets, and \(15\) lie only in \(B\). Someone calculates the number in “exactly one of the events” as \(12+7+15=34\). Correct the calculation using the diagram regions.

Hints

- Identify the regions whose elements have exactly one property. - Decide whether the overlap matches the phrase “exactly one.”

Solution

1. “Exactly one event” includes only the two nonoverlapping parts of the circles. 2. The \(7\) elements in the intersection satisfy both events and must not be counted. 3. The correct number is \(12+15=27\).

Answer

The correct number is \(27\); the \(7\) elements in the overlap are excluded.

Venn diagrams and counting 18

Diagram I shows four values placed in regions of sets \(A\) and \(B\). Diagram II has the same two circles, but the labels \(A\) and \(B\) have been switched. Suppose each value must keep the same set membership it has in Diagram I. Which values must switch sides in Diagram II, and which values can remain in the same relative region?

Hints

- In Diagram I, identify which value is only in \(A\) and which is only in \(B\). - Which regions depend on the name of one particular circle? - Which regions are unchanged when the two circle names are swapped?

Solution

1. Switching the circle labels changes the “only \(A\)” region into the “only \(B\)” region and vice versa. 2. Therefore, \(12\), which is only in \(A\), and \(9\), which is only in \(B\), must switch sides. 3. The intersection is still the intersection of the two sets, so \(7\) can remain in the overlap. 4. The outside-both region is unchanged, so \(4\) can remain outside both circles.

Answer

\(12\) and \(9\) must switch sides; \(7\) in the overlap and \(4\) outside both circles can remain in the same relative regions.

Venn diagrams and counting 19

A spinner can land on outcomes \(1\) through \(7\). The diagram separates outcomes in events \(A\) and \(B\). Nia says, “The union \(A\cup B\) includes only the overlapping outcomes, because that's where the two circles join.” Use the diagram to correct the claim and list the outcomes in \(A\cup B\).

Hints

- What does “in at least one of the two events” include besides the overlap? - Read the outcomes belonging only to \(A\) and only to \(B\). - Which outcome belongs to neither event?

Solution

1. The overlap contains \(3\) and \(4\). Those are the outcomes in \(A\cap B\), not the entire union. 2. Outcomes \(1,2\) are in \(A\) only, and \(5,6\) are in \(B\) only. All six outcomes are inside at least one circle. 3. Therefore, \(A\cup B=\{1,2,3,4,5,6\}\). Outcome \(7\) lies outside both circles.

Answer

Nia is incorrect. The overlap is \(A\cap B=\{3,4\}\); the union includes every outcome in either circle, \(A\cup B=\{1,2,3,4,5,6\}\).

Venn diagrams and counting 20

A costume shop has \(48\) orders. Of the orders, \(19\) include only a mask, \(8\) include both a mask and a cape, and \(13\) include only a cape. How many orders include exactly one item, at least one item, and neither item?

Hints

- Match each question to the appropriate Venn-diagram regions. - Use the total number of orders to find the outside region.

Solution

1. Exactly one item appears in \(19+13=32\) orders. 2. At least one item appears in \(19+8+13=40\) orders. 3. Neither item appears in \(48-40=8\) orders.

Answer

Exactly one: \(32\) orders At least one: \(40\) orders Neither: \(8\) orders

Venn diagrams and counting 21

The rectangle represents the sample space \(S\). Exactly one region is shaded. a) Write the shaded event using set notation. b) Describe the event in words.

Hints

- Identify which circle contains the shaded region. - Check whether the overlap is shaded or excluded. - Think about how “in one event but not the other” is written symbolically.

Solution

1. The shaded region is inside \(A\) but excludes every point that is also in \(B\). 2. That region is \(A\setminus B\), equivalently \(A\cap B^c\).

Answer

a) \(A\setminus B\), equivalently \(A\cap B^c\). b) Outcomes that are in \(A\) but not in \(B\).

Venn diagrams and counting 22

The rectangle represents the sample space \(S\). The diagram shows one shaded region. Write the shaded event in two equivalent forms: one using a complement of a union and one using an intersection of complements.

Hints

- First name the region consisting of everything inside either circle. - The shading is the complement of that entire region. - How does De Morgan's law rewrite the complement of a union?

Solution

1. The shaded outcomes are not in \(A\) or \(B\), so they are outside \(A\cup B\). 2. Therefore, the event is \((A\cup B)^c\). 3. By De Morgan's law, the same event is \(A^c\cap B^c\).

Answer

\((A\cup B)^c=A^c\cap B^c\).

Venn diagrams and counting 23

The rectangle represents the sample space \(S\). Use the nested Venn diagram to simplify each expression. a) \(A\cap B\) b) \(A\cup B\) c) \(B\setminus A\)

Hints

- What does it mean that one circle lies completely inside the other? - For the intersection, keep only outcomes in both sets. - For the set difference, ask whether the inner circle has any part outside the outer circle.

Solution

1. The diagram shows that every outcome in \(B\) is also in \(A\), so \(B\subseteq A\). 2. Therefore, \(A\cap B=B\). 3. Since \(B\) adds no outcomes outside \(A\), \(A\cup B=A\). 4. There are no outcomes in \(B\) that lie outside \(A\), so \(B\setminus A=\varnothing\).

Answer

a) \(B\) b) \(A\) c) \(\varnothing\)

Venn diagrams and counting 24

A school survey shows \(8\) students in club \(A\), \(7\) in club \(B\), \(2\) in both, and \(3\) in neither. Use the Venn diagram to give the number of students in each labeled region \(P\), \(Q\), \(R\), and \(S\).

Hints

- Which letter lies in both circles? - The club totals include the students in both clubs. - Find the region outside both circles last.

Solution

1. Region Q is the overlap, so Q has \(2\). 2. Region P is A only, so P has \(8-2=6\). 3. Region R is B only, so R has \(7-2=5\). 4. Region S is outside both, so S has \(3\).

Answer

P: \(6\); Q: \(2\); R: \(5\); S: \(3\).

Venn diagrams and counting 25

Each number in the Venn diagram is a count of students. a) How many students are in at least one of the two groups? b) How many students are not in group M?

Hints

- Which regions would be reached by someone in either group? - For part b), which parts lie beyond the boundary of \(M\)?

Solution

1. At least one group means M or N or both: \(8+4+11=23\). 2. Not in M includes the N-only region and the outside region: \(11+3=14\).

Answer

a) \(23\) b) \(14\)

Venn diagrams and counting 26

There are \(30\) students in total. The diagram shows how many are in each region, except for one missing number. Find the missing number and then find how many students are in \(A\cup B\).

Hints

- How many separate regions are there in the rectangle? - After finding the missing number, which regions belong to the event asked for?

Solution

1. The known regions have \(9+8+5=22\) students, leaving \(30-22=8\) in the shared region. 2. Inside either circle are \(9+8+8=25\) students.

Answer

Missing overlap: \(8\). In \(A\cup B\): \(25\) students.

Venn diagrams and counting 27

Amir looks at the Venn diagram and says \(20\) students belong to \(C\cup D\). Is his count correct? Explain which group he counted that does not belong in the union, and find the correct total.

Hints

- Which parts of the diagram lie inside at least one circle? - What might Amir have included that is outside both circles?

Solution

1. The union includes the students inside either circle: \(7+6+4=17\). 2. Amir counted all four regions, including the \(3\) students outside both circles, giving \(20\). 3. Those outside students do not belong to \(C\cup D\). The correct total is \(17\).

Answer

No. He included the \(3\) students in neither group. There are \(17\) students in \(C\cup D\).

Venn diagrams and counting 28

A class has \(24\) students. The diagram records who plays an instrument and who plays a school sport. The circle labels give each group's total. One student is chosen at random. a) What is the probability that the student does neither activity? b) What is the probability that the student plays an instrument but not a sport? c) Elif says, “The two activity totals add to the class size, so everyone must do at least one.” Explain her mistake.

Hints

- Find the shared students before working out either exclusive group. - How many students are inside at least one circle? - Does adding two circle totals always count different students?

Solution

1. The overlap contains \(6\) students. The instrument-only region contains \(14-6=8\); the sport-only region contains \(10-6=4\). 2. Outside both circles are \(24-(8+6+4)=6\) students. The probability in a) is \(\frac{6}{24}=\frac{1}{4}\). 3. The instrument-only probability is \(\frac{8}{24}=\frac{1}{3}\). 4. Elif's addition counts the same \(6\) students twice. Only \(18\) students do at least one activity.

Answer

a) \(\frac{1}{4}\). b) \(\frac{1}{3}\). c) The overlap is counted twice; \(6\) students do neither activity.

Venn diagrams and counting 29

The Venn diagram shows the sizes of a school's drama club and choir. A total of \(60\) students belong to at least one of the two clubs. Find how many belong to both clubs, drama only, and choir only.

Hints

- How does the sum of the two circle totals compare with the number in either circle? - Why would that difference equal the shared region? - After finding the shared students, what remains in each circle?

Solution

1. The circle totals are \(45\) for drama and \(32\) for choir. Their sum \(77\) counts shared members twice. 2. The overlap is \(77-60=17\). 3. Drama only is \(45-17=28\), and choir only is \(32-17=15\). 4. Check that the three inside regions add to \(28+17+15=60\).

Answer

Both: \(17\). Drama only: \(28\). Choir only: \(15\).

Venn diagrams and counting 30

Use the letters in the Venn diagram to name every region belonging to each event. a) \(A\cup B\) b) \(A\cap B\) c) \(A\cap B^c\) d) \(A^c\cap B^c\)

Hints

- Check where each letter lies relative to both circles. - The union means at least one, while the intersection means both. - A complement also includes parts of the rectangle outside a circle.

Solution

1. The union includes both exclusive circle regions and their overlap: \(W,X,Y\). 2. The intersection includes only the overlap, labeled \(X\). 3. The part in \(A\) and not \(B\) is \(W\). 4. The outside-both part of the sample space is \(Z\).

Answer

a) \(W,X,Y\). b) \(X\). c) \(W\). d) \(Z\).

Venn diagrams and counting 31

A Venn diagram has \(x\) elements only in \(A\), \(6\) elements in both sets, \(9\) elements only in \(B\), and \(4\) elements outside both sets. There are \(31\) elements in all. Find \(x\), and then find \(n(A)\).

Hints

- First use the sum of all four nonoverlapping regions. - Then identify which regions make up circle \(A\).

Solution

1. Add the four nonoverlapping regions: \(x+6+9+4=31\). 2. Solve for \(x\): \(x=31-19=12\). 3. Set \(A\) contains the “only \(A\)” region and the overlap. 4. Therefore, \(n(A)=12+6=18\).

Answer

\(x=12\), and \(n(A)=18\).

Venn diagrams and counting 32

A Venn diagram for two courses represents \(52\) students. Of these, \(21\) take course \(A\), \(26\) take course \(B\), and \(9\) take both. Find the number of students in each of the four diagram regions.

Hints

- Each course total already includes the students who take both courses. - Find the outside region only after determining the three regions inside the circles.

Solution

1. The number who take only course \(A\) is \(21-9=12\). 2. The overlap contains \(9\) students. 3. The number who take only course \(B\) is \(26-9=17\). 4. The circles contain \(12+9+17=38\) students. 5. The number outside both circles is \(52-38=14\).

Answer

Only \(A\): \(12\); both courses: \(9\); only \(B\): \(17\); neither course: \(14\).

Venn diagrams and counting 33

A Venn diagram shows \(n(A)=34\), \(23\) elements only in \(A\), \(16\) elements only in \(B\), and \(8\) elements outside both sets. Find the number in the overlap, \(n(B)\), and \(n(S)\).

Hints

- Use the parts of circle \(A\) to find the overlap. - Then build the other requested totals from the diagram regions.

Solution

1. The overlap is the part of \(A\) not included in the “only \(A\)” region: \(34-23=11\). 2. Set \(B\) contains its exclusive region and the overlap: \(n(B)=16+11=27\). 3. The union contains \(23+11+16=50\) elements. 4. Including the outside region gives \(n(S)=50+8=58\).

Answer

The overlap contains \(11\) elements, \(n(B)=27\), and \(n(S)=58\).

Venn diagrams and counting 34

A Venn diagram represents the events \(A\): “the number is divisible by \(2\)” and \(B\): “the number is divisible by \(4\)” for the sample space \(S=\{1, 2, 3, 4, 5, 6, 7, 8\}\). Describe how the two circles must be positioned, and place every number in the correct region.

Hints

- Determine whether one divisibility condition always implies the other. - Then test each number in the sample space.

Solution

1. Every number divisible by \(4\) is also divisible by \(2\), so circle \(B\) must lie entirely inside circle \(A\). 2. The elements in \(A\) but not \(B\) are \(2\) and \(6\). 3. The elements in \(B\) are \(4\) and \(8\). 4. The elements outside \(A\) are \(1, 3, 5, 7\). The region in \(B\) but not \(A\) is empty.

Answer

Circle \(B\) lies entirely inside circle \(A\). \(A\) only: \(\{2, 6\}\); \(B\): \(\{4, 8\}\); outside both: \(\{1, 3, 5, 7\}\); \(B\) only: \(\varnothing\).

Venn diagrams and counting 35

A transit agency reviews \(500\) monthly passes. Of the passes, \(420\) are valid on buses, represented by \(B\), \(310\) are valid on commuter trains, represented by \(T\), and \(260\) are valid on both. What percent of the passes are valid on exactly one of the two forms of transportation?

Hints

- Find each nonoverlapping region separately. - Divide their combined count by the total number of passes.

Solution

1. The number valid only on buses is \(420-260=160\). 2. The number valid only on commuter trains is \(310-260=50\). 3. The number valid on exactly one is \(160+50=210\). 4. The percentage is \(\frac{210}{500}\cdot100\%=42\%\).

Answer

\(210\) passes, or \(42\%\), are valid on exactly one form of transportation.

Venn diagrams and counting 36

At a community repair workshop, \(75\) devices are inspected. Of the devices, \(28\) have only a mechanical defect, \(19\) have only an electrical defect, and \(11\) have both. What is the probability that a randomly selected device has neither defect? Give the result as a fraction and as a percent rounded to the nearest tenth.

Hints

- Add the three regions inside the two sets. - Subtract from the total to find the outside region.

Solution

1. The number with at least one defect is \(28+19+11=58\). 2. The number with neither defect is \(75-58=17\). 3. The probability is \(\frac{17}{75}\approx0.2267\). 4. As a percent rounded to the nearest tenth, this is \(22.7\%\).

Answer

\(\frac{17}{75}\approx22.7\%\)

Venn diagrams and counting 37

A coding club reviews \(100\) projects. Let \(A\) be the event that a project uses a microcontroller, and let \(B\) be the event that it uses a web interface. Suppose \(P(A)=0.58\), \(P(B)=0.46\), and \(P(A\cap B)=0.27\). Find the number of projects in each of the four regions of a two-set Venn diagram.

Hints

- With a total of \(100\), convert each probability directly to a count. - Subtract the overlap from each individual event count.

Solution

1. Because there are \(100\) projects, the probabilities correspond directly to counts: \(58\) use a microcontroller, \(46\) use a web interface, and \(27\) use both. 2. The count in \(A\) only is \(58-27=31\). 3. The count in \(B\) only is \(46-27=19\). 4. The union count is \(31+27+19=77\). 5. The count in neither event is \(100-77=23\).

Answer

\(A\) only: \(31\) Both: \(27\) \(B\) only: \(19\) Neither: \(23\)

Venn diagrams and counting 38

An audiobook app classifies \(160\) titles by two characteristics. Let \(A\) be the event that a title is shorter than \(5\) hours, and let \(B\) be the event that it is nonfiction. Of the titles, \(74\) are shorter than \(5\) hours, \(58\) are nonfiction, and \(103\) satisfy at least one characteristic. Find the number of titles in each of the four regions of a two-set Venn diagram.

Hints

- Use the individual counts and union count to find the overlap. - Then find each remaining region by subtraction.

Solution

1. The overlap count is \(n(A\cap B)=74+58-103=29\). 2. The count in \(A\) only is \(74-29=45\). 3. The count in \(B\) only is \(58-29=29\). 4. The count in neither event is \(160-103=57\).

Answer

\(A\) only: \(45\) Both: \(29\) \(B\) only: \(29\) Neither: \(57\)

Venn diagrams and counting 39

The Venn diagram shows counts in the four regions for events \(A\) and \(B\). The overlap is labeled \(x\). You are also told that \(n(A)=25\). a) Find \(x\). b) Find \(n(S)\). c) A classmate claims that \(n(B)=31\). Is the claim consistent with the diagram? Explain.

Hints

- Which two regions together make event \(A\)? - After finding the overlap, include every region once to obtain \(n(S)\). - For the claim about \(B\), use only the regions inside circle \(B\).

Solution

1. Event \(A\) consists of its only-\(A\) region and the overlap, so \(12+x=25\). Thus, \(x=13\). 2. The sample-space total is \(12+13+17+8=50\). 3. Event \(B\) contains the overlap and the \(B\)-only region, so \(n(B)=13+17=30\). Therefore, the claim \(n(B)=31\) is inconsistent with the diagram.

Answer

a) \(x=13\) b) \(n(S)=50\) c) No. The diagram gives \(n(B)=30\), not \(31\).

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Venn diagrams and counting

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