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Dành cho nhà giáo dục: slide bài học, ghi chú ôn tập sẵn dùng cho Sample spaces, tables and Venn diagrams (Mathematics, Probability) — dùng trong bài giảng của bạn, hoặc chạy chủ đề như một hoạt động lớp học tương tác để người học chơi như một trò chơi trực tiếp.
A sample space lists all possible outcomes of an experiment or combined experiment.
For a single event, list outcomes systematically, e.g. rolling a fair die: {1, 2, 3, 4, 5, 6}.
For two events, use a sample space diagram (a grid) to show every possible pair of outcomes.
Example: tossing two coins – outcomes: HH, HT, TH, TT.
A sample space can contain unequal-probability outcomes. Counting favourable outcomes over total works only when the listed outcomes are equally likely.
For equally likely elementary outcomes, probability = favourable outcomes ÷ total outcomes. Otherwise use their actual probabilities.
Sums of two independent fair dice
Two-Way Tables
A two-way table organises outcomes of two events in rows and columns.
Each cell shows the combined outcome or its count.
To find a probability from a two-way table, read the relevant cell(s) and divide by the grand total.
Example: a table of students by gender and subject choice – the cell for 'female and Maths' gives that count.
Always check that row totals and column totals add up to the grand total.
Two-way survey table
Venn Diagrams
A Venn diagram uses circles to represent sets inside a rectangle (the universal set, often denoted U).
Overlapping circles show the intersection – elements in both sets.
The union is all elements in either set or both.
Fill the intersection first, then work outwards to avoid double-counting.
Remember to include elements that are in U but outside all circles – they belong to neither set.
Intersection of A and B
Set Notation
A \cap B means the intersection of sets A and B – elements in both.
A \cup B means the union – elements in A or B or both.
A' (or \bar{A}) means the complement of A – elements not in A.
\emptyset is the empty set – no elements.
n(A) means the number of elements in set A.
Do not confuse \cap (intersection) with \cup (union) – remember \cap looks like an 'n' for 'and'.
Complement inside the universal set
Calculating Probabilities from Venn Diagrams
Once a Venn diagram is complete, probabilities can be read directly.
P(A \cap B) = \frac{n(A \cap B)}{n(U)}.
P(A \cup B) = \frac{n(A \cup B)}{n(U)}.
P(A') = 1 - P(A).
For 'neither A nor B', use the region outside both circles: P((A \cup B)') = 1 - P(A \cup B).
Probability of a union
Common Misconceptions
Double-counting the intersection when finding a union – subtract n(A \cap B) once.
Starting with the known overlap is helpful when set totals are given. Other mathematically valid solution orders are possible.
Forgetting elements outside all sets but inside the universal set – they are part of U and must be included.
Confusing \cap and \cup symbols – check the meaning.
Listing outcomes unsystematically can miss cases – use a table or grid to be systematic.
Typical Question Forms
Complete a two-way table from given information and find a probability.
Complete a Venn diagram from given information (e.g., totals and intersection).
Find a probability using set notation like P(A \cup B) or P(A').
List all outcomes for two combined events (e.g., rolling a die and tossing a coin).
Find how many elements satisfy neither condition – use the complement region.
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10.Match each probability to its value when two fair dice are rolled.
Medium
P(sum = 7)
P(sum = 12)
P(sum = 3)
6/36
1/36
2/36
11.Put the following steps in order to calculate the probability of the union of two events using a Venn diagram.
Medium
Identify the number of elements in each set.
Subtract the number in the intersection to avoid double-counting.
Add the numbers in both sets.
Divide by the total number of elements in the universal set.
12.In a Venn diagram, which symbol represents the union of two sets?
Easy
A∪
B∩
C⊂
D∅
13.A fair coin is tossed and a fair six-sided dice is rolled. How many outcomes are in the sample space?
Easy
A12
B8
C6
D36
14.The intersection of two sets A and B is the set of elements that are in both A and B.
Easy
True or false?
15.A Venn diagram shows 20 students: 12 study French, 10 study Spanish, and 4 study both. What is the probability that a randomly chosen student studies neither language?
Medium
A0.2
B0.1
C0.3
D0.4
16.Two fair dice are rolled. What is the probability that the sum of the two numbers is 7?
Medium
A1/6
B1/12
C1/9
D5/36
17.A bag contains 5 red and 3 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles are red?
Hard
A5/14
B25/64
C5/8
D15/56
18.Which of the following statements about Venn diagrams are true? (Select all that apply)
Medium
AThe intersection of two sets is the region where both sets overlap.
BThe union of two sets includes all elements in either set.
CThe complement of a set includes elements outside the universal set.
DWhen completing a Venn diagram, fill the intersection first.
EThe total number of elements in the universal set equals the sum of the numbers in all regions.
19.In a Venn diagram, the union of two sets is represented by the region inside both circles but not the overlap.
Easy
True or false?
20.The sample space of an experiment is the set of all possible outcomes.
Easy
True or false?
21.In set notation, what does A ∩ B represent?
Easy
AThe union of sets A and B
BThe intersection of sets A and B
CThe complement of set A
DThe empty set
22.Which representations show all 36 ordered die-pair outcomes when two dice are rolled? Select all that apply.
Medium
AA 6 by 6 grid containing every ordered pair
BA list of every pair except (6, 6)
CA 6 by 6 sum table with each row and column labelled by its die result
DA list containing only the 11 distinct sums from 2 to 12
23.In a Venn diagram showing two overlapping sets A and B, which regions make up the union A ∪ B? (Select all that apply)
Medium
AThe region inside A only
BThe region inside B only
CThe overlap of A and B
DThe region outside both A and B
24.Put the following steps in order to find the probability of the union of two events using a Venn diagram.
Medium
Fill in the intersection with the number of elements in both events.
Work out the number of elements in each of the outer regions.
Add the A-only, intersection and B-only counts, each exactly once.
Divide the union by the total number of elements in the universal set.
25.Which of the following are correct ways to list the sample space for tossing a coin and rolling a fair six-sided dice? (Select all that apply)
26.A two-way table shows the results of rolling two fair dice. What is the probability that the sum is 7?
Medium
A\frac{1}{6}
B\frac{1}{12}
C\frac{7}{36}
D5/36
27.Of 40 students, 18 play a sport and 10 both play a sport and an instrument. What is the probability that a randomly chosen student plays a sport but not an instrument?
Medium
A\frac{8}{40}
B\frac{18}{40}
C\frac{10}{40}
D\frac{28}{40}
28.A fair coin is tossed and a fair six-sided dice is rolled. What is the probability of getting a head and a number greater than 4?
Medium
A1/6
B1/3
C1/12
D1/2
29.A Venn diagram shows two sets A and B. Which of the following statements are true? (Select all that apply)
Medium
AThe intersection A ∩ B is the region inside both A and B.
BThe union A ∪ B is the region inside A or B or both.
CThe complement of A includes B-only and neither, within the universal set.
DThe complement of A is the region outside A, excluding B.
30.In a Venn diagram, the region outside both sets A and B but inside the universal set represents elements that are in neither A nor B.
Easy
True or false?
31.Put the following steps in order to complete a Venn diagram from given information.
Medium
Fill in the intersection region.
Fill in the regions for each set, excluding the intersection.
Fill in the region outside all sets but inside the universal set.
Check that the numbers in each circle add up to the given total.
32.A class of 30 students: 18 play football, 15 play tennis, 8 play both. What is the probability that a randomly chosen student plays neither sport?
Hard
A1/6
B1/3
C1/2
D1/4
33.A two-way table shows the outcomes when two fair dice are rolled. Which of the following statements are correct? (Select all that apply)
Medium
AThere are 36 equally likely outcomes.
BThe probability of a double is 1/6.
CThe probability of a sum of 7 is 1/6.
DThe probability of a sum less than 5 is 1/6.
34.A fair coin is tossed and a fair six-sided dice is rolled. What is the probability of getting a tail and a prime number on the dice?
Medium
A1/4
B1/6
C1/3
D1/2
35.A two-way table shows the sum of two fair dice. What is the probability that the sum is greater than 8?
Medium
A5/12
B5/36
C10/36
D1/3
36.A Venn diagram shows 40 students: 20 study Biology, 25 study Chemistry, and 10 study both. Which of the following statements are correct? (Select all that apply)
Medium
AThe number who study Biology only is 10.
BThe number who study Chemistry only is 15.
CThe number who study at least one of the two subjects is 35.
DThe number who study neither subject is 5.
37.In a Venn diagram showing two overlapping sets A and B, which region represents elements that are in A but not in B?
Easy
AThe intersection region
BThe region inside A only
CThe region inside B only
DThe region outside both circles
38.Which of the following is the correct set notation for the intersection of sets A and B?
Easy
AA ∪ B
BA ∩ B
CA' ∩ B
DA ∪ B'
39.A two-way table shows the results of rolling two dice. How many outcomes give a sum of 7?
Medium
40.In a class of 30 students, 18 play football, 15 play tennis, and 8 play both. How many students play neither sport?
Medium
41.A two-way table shows the outcomes when two fair dice are rolled. How many outcomes give a sum of 7?
Medium
42.Write the set notation for the intersection of sets A and B.
Medium
43.Which symbol represents the union of sets A and B?
Easy
A∪
B∩
C∈
D⊂
44.A Venn diagram shows 20 students: 12 study French, 10 study Spanish, and 4 study both. How many study neither language?
Hard
45.A two-way table shows the results of rolling two fair dice. How many outcomes give a sum of 7?
Medium
46.Complete the sentence using the correct set notation symbols.
Easy
The ____ of two sets A and B is the set of elements that are in both sets, written as ____. The ____ is the set of elements in either set, written as ____. The ____ of set A is the set of elements not in A.
47.Write the set notation for the complement of set A.
Medium
48.Complete the Venn diagram description: When completing a Venn diagram, you should fill the ___ region first, then work outwards. The region outside all sets but inside the universal set represents elements that are in ___ of the sets.
Medium
When completing a Venn diagram, you should fill the ____ region first, then work outwards. The region outside all sets but inside the universal set represents elements that are in ____ of the sets.
49.In a Venn diagram, 20 students are shown: 12 study French, 10 study Spanish, and 4 study both. How many students study neither language?
Hard
50.A fair coin is tossed three times. How many outcomes are in the sample space?
Medium
51.Explain why, when finding the probability of the union of two events, you must subtract the probability of their intersection.
Hard
52.A bag contains 5 red and 3 blue marbles. Two marbles are drawn at random without replacement. Explain how to use a sample space diagram to find the probability that both marbles are the same colour. Calculate this probability. Treat the individual marbles as equally likely, and explain why the four colour sequences are not equally likely.
Hard
53.A bag contains 5 red and 3 blue marbles. Two marbles are drawn at random without replacement. Explain how to use a sample space diagram to find the probability that the two marbles are different colours. Treat the individual marbles as equally likely, and explain why the four colour sequences are not equally likely.
Hard
54.Explain why, when using a Venn diagram to find the probability of the union of two events, you must subtract the probability of their intersection.
Hard
55.A bag contains 5 red and 3 blue marbles. Two marbles are drawn at random without replacement. Explain how to use a sample space diagram to find the probability that both marbles are red. Treat the individual marbles as equally likely, and explain why the four colour sequences are not equally likely.
Hard
56.A two-way table shows the results of rolling two dice. How many outcomes give a sum of 7?
Medium
A8
B5
C7
D6
57.A two-way table shows the outcomes when two fair dice are rolled. How many outcomes give a sum of 7?
Medium
A8
B5
C7
D6
58.A Venn diagram shows 20 students: 12 study French, 10 study Spanish, and 4 study both. How many study neither language?
Hard
A3
B1
C2
D4
59.A two-way table shows the results of rolling two fair dice. How many outcomes give a sum of 7?
Medium
A7
B6
C8
D5
60.In a Venn diagram, 20 students are shown: 12 study French, 10 study Spanish, and 4 study both. How many students study neither language?
Hard
A3
B1
C2
D4
61.A fair coin is tossed three times. How many outcomes are in the sample space?
Medium
A9
B8
C10
D7
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