D. Data · Two-Variable Data, Scatter Plots and Probability

Probability with Venn Diagrams: And, Or, Not

D2.1About 50 minutes

Learning goals

I can organize outcomes in a Venn diagram.
I can find the probability of A and B, A or B, and not A.
I can use P(A or B) = P(A) + P(B) − P(A and B) and explain why the overlap is subtracted.
I can tell when two events are mutually exclusive.

Words to know

event
A set of one or more outcomes, like "rolling an even number".

A and B
Both events happen. In a Venn diagram, it is the overlap of the two circles.

A or B
At least one of the events happens (A, or B, or both). In a Venn diagram, it is everything inside either circle.

complement (not A)
All the outcomes where A does not happen. P(not A) = 1 − P(A).

mutually exclusive
Two events that cannot happen at the same time, so their circles don't overlap. Example: rolling a 1 and rolling a 6 on one roll.

Remember

Probability = number of favourable outcomes ÷ total number of equally likely outcomes. It goes from 0 (impossible) to 1 (certain), and can be written as a fraction, decimal or percent. You may also remember Venn diagrams from sorting in earlier grades. Now they become a probability tool.

Hockey, soccer, both or neither?

In class 8B there are 30 students. 18 play hockey, 12 play soccer, and 7 of them play both. Seven students play neither.

If you add 18 + 12, you get 30, which seems to be the whole class. But that's wrong! The 7 students who play both were counted twice. A Venn diagram sorts this out.

The rectangle is everyone: all 30 students.
Each circle is one event.
The overlap holds students in both circles.
Outside the circles, but inside the rectangle, are students in neither.

Hockey only: 18 − 7 = 11. Both: 7. Soccer only: 12 − 7 = 5. Neither: 7 (in the bottom-right corner, outside both circles). Check: 11 + 7 + 5 + 7 = 30.

And, or, not

Choose one student from 8B at random.

P(hockey and soccer) = the overlap = 7/30.
P(hockey or soccer) = everything inside the circles = 11 + 7 + 5 = 23, so 23/30.
P(not hockey) = everyone outside the hockey circle = 5 + 7 = 12, so 12/30 = 2/5. Check: 1 − 18/30 = 12/30. ✓

The addition rule

P(hockey) + P(soccer) = 18/30 + 12/30 = 30/30, which counts the overlap twice. Subtract it once:

P(A or B) = P(A) + P(B) − P(A and B)

18/30 + 12/30 − 7/30 = 23/30. ✓

If two events are mutually exclusive (they can't both happen), the overlap is empty, so P(A or B) = P(A) + P(B).

Worked example

Example 1: A number Venn diagram

A number from 1 to 20 is chosen at random. Event A: it is a multiple of 2. Event B: it is a multiple of 3. Find P(A and B), P(A or B) and P(neither).

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 (10 numbers).
Multiples of 3: 3, 6, 9, 12, 15, 18 (6 numbers).
Both: 6, 12, 18 (3 numbers). These go in the overlap.
Only multiples of 2: 2, 4, 8, 10, 14, 16, 20 (7). Only multiples of 3: 3, 9, 15 (3). Neither: 1, 5, 7, 11, 13, 17, 19 (7). Check: 7 + 3 + 3 + 7 = 20. ✓
P(A and B) = 3/20. P(A or B) = 10/20 + 6/20 − 3/20 = 13/20. P(neither) = 7/20.

P(A and B) = 3/20, P(A or B) = 13/20 = 65%, P(neither) = 7/20 = 35%. Notice that 13/20 + 7/20 = 1: "A or B" and "neither" are complements.

Numbers 1 to 20: 7 multiples of 2 only, 3 in both (6, 12, 18), 3 multiples of 3 only, and 7 in neither.

Worked example

Example 2: A deck of cards

A standard deck has 52 cards: 26 red and 26 black, and 12 face cards (jacks, queens and kings), of which 6 are red. One card is drawn at random. Find P(red or face card).

P(red) = 26/52. P(face) = 12/52. P(red and face) = 6/52.
Use the addition rule: 26/52 + 12/52 − 6/52 = 32/52.
Simplify: 32/52 = 8/13 ≈ 0.615 ≈ 61.5%.

P(red or face) = 8/13 (about 62%). Without subtracting the 6 red face cards, you'd get 38/52, which counts them twice.

Worked example

Example 3: Mutually exclusive or not?

Roll one standard die. (a) Are "roll a 1" and "roll a 6" mutually exclusive? Find P(1 or 6). (b) Are "roll an even number" and "roll a number greater than 3" mutually exclusive? Find P(even or greater than 3).

(a) One roll can't be both 1 and 6, so they're mutually exclusive. P(1 or 6) = 1/6 + 1/6 = 2/6 = 1/3.
(b) Even: 2, 4, 6. Greater than 3: 4, 5, 6. Both: 4 and 6. So they are not mutually exclusive.
P(even or greater than 3) = 3/6 + 3/6 − 2/6 = 4/6 = 2/3. Check by listing: 2, 4, 5, 6 is 4 outcomes. ✓

(a) Mutually exclusive, 1/3. (b) Not mutually exclusive, 2/3.

Strategy

Draw the rectangle and circles. Fill in the overlap first.
Subtract to get the "only" parts. Then find "neither".
Check that all four regions add to the total.
Read the probability you need: overlap for and, everything in the circles for or, everything outside a circle for not.

Just for fun

Why do the two circles in a Venn diagram get along so well?

They have a lot in common. In 8B, exactly 7 students' worth.

Hands-on activity

Card-draw experiment

You need: a standard deck of 52 playing cards, paper and pencil.

Draw a Venn diagram with circles for Red and Face card. Using the deck, count the cards in each region: red face cards, red non-face cards, black face cards, and black non-face cards.
Find the theoretical P(red or face card). (It should be 32/52 = 8/13.)
Shuffle, draw one card, record whether it is red, a face card, both or neither, and put it back. Shuffle again. Do 40 draws.
Find your experimental P(red or face) = (number of draws that were red or a face card) ÷ 40. Compare it with 8/13 ≈ 62%.

What to look for: the four regions are 6 (red face), 20 (red only), 6 (black face) and 20 (neither). Your experiment should be somewhere near 62%, but it won't be exact with only 40 draws.

Your turn

Use the 8B Venn diagram. What is P(soccer only)?

Use the 8B Venn diagram. What is P(neither sport)?

A survey of 40 people: 25 have a dog, 15 have a cat, 6 have both. How many have a dog only? A cat only? Neither?

In the pet survey, find P(dog or cat).

Are "draw a heart" and "draw a spade" (one card from a deck) mutually exclusive? Explain.

One die is rolled. Find P(odd or less than 3).

If P(snow tomorrow) = 0.35, what is P(no snow tomorrow)?

More practice: warm-up

In a Venn diagram, where do you find "A and B"?

If P(A) = 3/10, what is P(not A)?

In 8B, how many students play hockey only?

Roll a die. P(2 or 5)?

What does the rectangle in a Venn diagram stand for?

More practice: on your own

In a club of 50 students, 28 play an instrument, 20 sing in the choir, and 9 do both. Find P(instrument or choir) for a randomly chosen student.

In the same club, find P(neither).

A number from 1 to 30 is chosen at random. Find P(multiple of 5 or multiple of 6).

One card is drawn from a standard deck. Find P(king or heart).

Explain why P(hockey) + P(soccer) = 30/30 does not mean every student in 8B plays a sport.

Challenge

In a group of 60 students, 35 take French immersion, 22 play in the band, and 12 do neither. How many do both?

P(A) = 0.5, P(B) = 0.4 and P(A or B) = 0.7. Find P(A and B). Are A and B mutually exclusive?

Draw (in words) a Venn diagram with three circles for a number from 1 to 12: multiples of 2, multiples of 3 and multiples of 4. What number(s) go in the middle where all three overlap? Find P(all three).

Check your understanding

Quiz

In 8B, 18 play hockey, 12 play soccer and 7 play both (30 students). What is P(hockey or soccer)?

In a Venn diagram, "A and B" is...

Which pair of events is mutually exclusive for one roll of a die?

If P(A) = 0.42, what is P(not A)?

One card from a standard deck. P(red or face card) = ?

Why do we subtract P(A and B) in the addition rule?

For parents and teachers

Big idea: The Ontario curriculum (D2.1) asks Grade 8 students to solve various probability problems using appropriate tools and strategies, including Venn and tree diagrams. This lesson covers Venn diagrams; the next two lessons use tree diagrams.

Common difficulties:

Double-counting the overlap (adding 18 + 12 without subtracting the 7 who do both).
Writing the circle totals in the circles instead of the "only" numbers: fill the overlap first, then subtract.
Forgetting the "neither" region.
Mixing up "and" (overlap) and "or" (everything in the circles).

How to help: Use real lists: family members who like pizza and who like sushi. Draw a quick Venn diagram on a napkin and ask "and", "or" and "not" questions. A deck of cards is a great tool for this lesson.