A. Counting and Probability · Probability

Venn Diagrams: Intersection, Union and Counting

A1.5About 70 minutes

Learning goals

I can describe the intersection A ∩ B and the union A ∪ B of two events and shade them in a Venn diagram.
I can fill in a two-set or three-set Venn diagram from given counts, working from the centre out.
I can use the principle of inclusion and exclusion, n(A ∪ B) = n(A) + n(B) − n(A ∩ B), and extend it to three sets.
I can calculate probabilities of compound events from a Venn diagram.

Words to know

intersection (A ∩ B)
The event "A and B": the outcomes that are in both A and B.

union (A ∪ B)
The event "A or B": the outcomes in A, in B, or in both. In mathematics, "or" includes "both".

Venn diagram
A diagram in which a rectangle represents the sample space S and circles represent events; overlapping regions show intersections.

principle of inclusion and exclusion
n(A ∪ B) = n(A) + n(B) − n(A ∩ B): add the sets, then subtract what was counted twice.

region
One of the separate areas of a Venn diagram, such as "in A but not in B" (A ∩ B′).

And, or, and the overlap

In the last lesson you met the complement A′. Two more set operations combine events:

Intersection A ∩ B ("A and B"): outcomes in both events.
Union A ∪ B ("A or B"): outcomes in at least one of the events. In math, "or" is inclusive: an outcome in both events still counts.

For a die roll with A = {2, 4, 6} (even) and B = {4, 5, 6} (greater than 3):

A ∩ B = {4, 6}
A ∪ B = {2, 4, 5, 6}
A ∩ B′ = {2} (even but not greater than 3)

A Venn diagram shows this at a glance. The rectangle is S, each event is a circle, and the overlap is the intersection. Two overlapping circles make 4 regions: A only, both, B only, and neither.

The shaded area is A ∪ B: everything in A, in B, or in both. Each of the four regions has its own name; the unshaded region outside both circles is (A ∪ B)′, "neither".

Counting with inclusion and exclusion

If you add n(A) and n(B), the outcomes in the overlap get counted twice. Subtract them once:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Dividing by n(S) gives the same rule for probabilities:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

You used this formula in Grade 8. In lesson 9 you will see when the overlap is empty (mutually exclusive events) and the formula simplifies.

Worked example

Example 1: Filling a two-set Venn diagram

In a survey of 120 Grade 12 students, 68 play a school sport, 45 are in a music program, and 20 do both. (a) Draw a Venn diagram. (b) How many students do neither? (c) Find P(a randomly chosen student from this group plays a sport or is in music). (d) Find P(sport only).

Let T = plays a sport and M = in music. Start with the overlap: T ∩ M = 20.
Sport only: 68 − 20 = 48. Music only: 45 − 20 = 25.
In at least one: 48 + 20 + 25 = 93. Check with the formula: 68 + 45 − 20 = 93. ✓
(b) Neither: 120 − 93 = 27.
(c) P(T ∪ M) = 93/120 = 31/40 = 0.775.
(d) P(T ∩ M′) = 48/120 = 2/5 = 0.4.

(b) 27 students; (c) 31/40; (d) 2/5.

The completed Venn diagram for Example 1. The four regions (48, 20, 25 and 27) add to 120, the whole group.

Work from the inside out

Always fill the innermost region (the overlap) first, then subtract to get the "only" regions, then subtract from the total to get "neither". A common error is to write 68 in the sport-only region, which counts the 20 students in both twice.

Three events

Three overlapping circles make 8 regions: the centre (all three), three "exactly two" regions, three "only one" regions, and the outside. The counting rule extends:

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)

Why add the last term back? Outcomes in all three sets are added 3 times and then subtracted 3 times, so they would be missed entirely without adding them once more.

Worked example

Example 2: A three-set survey

A school surveyed 100 students about winter activities this year: 45 went skating (K), 30 went skiing (I), 25 went snowshoeing (W). 12 did skating and skiing, 10 did skating and snowshoeing, 8 did skiing and snowshoeing, and 4 did all three. (a) Complete a Venn diagram. (b) How many did none of the three? (c) Find P(exactly one activity).

Centre: K ∩ I ∩ W = 4.
Exactly two (subtract the centre): K and I only = 12 − 4 = 8; K and W only = 10 − 4 = 6; I and W only = 8 − 4 = 4.
Only one: skating only = 45 − 8 − 6 − 4 = 27; skiing only = 30 − 8 − 4 − 4 = 14; snowshoeing only = 25 − 6 − 4 − 4 = 11.
Union: 27 + 14 + 11 + 8 + 6 + 4 + 4 = 74. Check: 45 + 30 + 25 − 12 − 10 − 8 + 4 = 74. ✓
(b) None: 100 − 74 = 26.
(c) Exactly one: 27 + 14 + 11 = 52, so P = 52/100 = 0.52.

(b) 26 students; (c) P(exactly one) = 0.52.

The completed three-set Venn diagram for Example 2. The eight numbers add to 100. The centre (4) did all three activities.

Worked example

Example 3: Working backward

In a group of 50 people, 30 have a library card and 28 have a transit pass. 7 have neither. How many have both?

In at least one: 50 − 7 = 43.
Use n(L ∪ T) = n(L) + n(T) − n(L ∩ T): 43 = 30 + 28 − n(L ∩ T).
n(L ∩ T) = 58 − 43 = 15.

15 people have both.

Just for fun

Riddle: Why did the student refuse to sit in the middle region of the Venn diagram?

Too much pressure from all sides: three circles all claiming the same seat.

Hands-on activity

Venn diagram of your household

You need: a large sheet of paper, three pieces of string or yarn (or draw circles), and some sticky notes or small paper slips.

Choose three properties for people you know (family, friends, neighbours), for example "likes hockey", "has been to Québec", "plays an instrument". Lay out three overlapping loops of string and label them.
Write each person's name on a slip and place it in the correct region. Anyone with none of the properties goes outside the loops but inside the paper (the sample space).
Count each of the 8 regions.
Check the three-set inclusion-exclusion formula with your numbers.
If you pick one of these people at random, what is P(at least two properties)?

What to look for: every name is in exactly one region; the region counts add to the total number of people; the formula's answer matches the number of names inside at least one loop. (Keep it friendly: choose properties people are happy to share.)

Your turn

S = {1, 2, …, 12}, A = {multiples of 2}, B = {multiples of 3}. List A ∩ B and A ∪ B.

Using the sets in the previous question, find P(A ∪ B) using the formula and check by counting.

n(A) = 40, n(B) = 25, n(A ∩ B) = 10. Find n(A ∪ B).

In a class of 30, 18 have a part-time job, 12 have a driver's licence, and 5 have both. How many have neither?

Using the class in the previous question, find P(job but no licence).

Describe in words the region A′ ∩ B.

P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7. Find P(A ∩ B).

More practice: warm-up

Which symbol means "and"?

Which symbol means "or"?

How many regions do two overlapping circles make inside S?

How many regions do three overlapping circles make inside S?

A = {a, b, c}, B = {c, d}. A ∩ B?

More practice: on your own

One card is drawn. Let R = red and F = face card. Find n(R ∩ F), n(R ∪ F) and P(R ∪ F).

Of 200 households, 150 have internet streaming, 90 have a game console, and 20 have neither. How many have both?

Show with a Venn diagram (in words) why n(A ∪ B) ≤ n(A) + n(B).

In a three-set problem, n(A ∩ B ∩ C) = 3 and n(A ∩ B) = 10. How many are in A and B but not C?

P(A) = 0.45, P(B) = 0.35, P(A ∩ B) = 0.15. Find P(neither A nor B).

Challenge

How many whole numbers from 1 to 100 are divisible by 2, 3 or 5?

In a group of 60 students, 35 study French, 28 study Spanish and 20 study Mandarin. 12 study French and Spanish, 9 French and Mandarin, 7 Spanish and Mandarin, and 4 all three. Find P(a student studies exactly two of the languages).

P(A) = 0.6 and P(B) = 0.7. What are the smallest and largest possible values of P(A ∩ B)?

Check your understanding

Quiz

S = {1, 2, …, 10}, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}. What is A ∪ B?

n(A) = 18, n(B) = 15 and n(A ∪ B) = 27. What is n(A ∩ B)?

P(A) = 0.3, P(B) = 0.5, P(A ∩ B) = 0.1. What is P(A ∪ B)?

In a Venn diagram for 40 people, 25 like tea, 20 like coffee and 10 like both. How many like tea only?

Why is n(A ∩ B ∩ C) added back in the three-set inclusion and exclusion formula?

Which region is described by A ∩ B′?

For parents and teachers

Big idea (A1.5, Venn diagrams): intersection (∩, "and") is the overlap; union (∪, "or") is everything in either set. Counting a union means adding and then subtracting the overlap that was counted twice.

How to help: The household Venn activity is a hands-on way to see the regions. When your teen fills a Venn diagram, ask "Which number goes in first?" (the centre).

Common difficulties:

Putting the total for a set (like 68) in the "only" region.
Forgetting the "neither" region outside the circles.
Reading "or" as "one or the other but not both".
In three-set problems, forgetting to subtract the centre from each pair before filling the "exactly two" regions.