Sample spaces, Venn diagrams and tree diagrams

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Sample Spaces

A sample space is the set of all possible outcomes of an experiment or random trial.
It is usually written using set notation, with outcomes listed as elements inside curly brackets, e.g. S = {H, T} for a coin toss.
The sample space may be denoted by S, Ω, or U (for 'universal set').
The outcomes can be numbers, words, letters, or symbols.
A sample space can be finite, countably infinite, or uncountably infinite.
For a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
For two coins, the sample space is {HH, HT, TH, TT}, where the order matters (first coin, then second).

Events

An event is a subset of the sample space, usually denoted by E.
If the outcome of an experiment is in E, then event E has occurred.
For a single coin toss, possible events include E = {} (empty set), E = {H}, E = {T}, and E = {H, T}.
For two coins, the event 'at least one head' is E = {HH, HT, TH}.
Events can be shown visually: a rectangle represents the sample space, and ovals inside represent events.

Conditions for a Sample Space

Outcomes must be mutually exclusive: if one outcome occurs, no other outcome can occur at the same time.
Outcomes must be collectively exhaustive: on every trial, one of the outcomes in the sample space must occur.
A well-defined, non-empty sample space S is one component of a probabilistic model (a probability space).
The other components are a set of possible events and a probability assigned to each event.

Listing Outcomes Systematically

To find a sample space, list all possible outcomes in an organised way so none are missed.
For two events, use a sample space diagram (a grid) to show all combinations.
Example: rolling two dice, list pairs (1,1), (1,2), …, (6,6) — there are 36 outcomes.
Order matters when the events are distinguishable (e.g. first die and second die).

Venn Diagrams and Set Notation

A Venn diagram uses circles inside a rectangle to show sets and their relationships.
The rectangle represents the universal set (the sample space).
Union (A ∪ B) means 'A or B' — outcomes in A, in B, or in both.
Intersection (A ∩ B) means 'A and B' — outcomes in both A and B.
Complement (A') means 'not A' — outcomes in the universal set but not in A.
Venn diagrams help visualise events and their probabilities.

Mutually Exclusive Events and the Addition Rule

Mutually exclusive events cannot happen at the same time; their intersection is empty.
For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
This is the addition rule for mutually exclusive events.
If events are not mutually exclusive, P(A or B) = P(A) + P(B) − P(A and B).

Independent Events and the Multiplication Rule

Independent events are events where the outcome of one does not affect the other.
For independent events A and B, P(A and B) = P(A) × P(B).
This is the multiplication rule for independent events.
Example: tossing two coins, P(two heads) = 1/2 × 1/2 = 1/4.

Tree Diagrams

A tree diagram shows all possible outcomes of two or more events in a branching structure.
Each branch represents a possible outcome, with its probability written on the branch.
To find the probability of a combination, multiply the probabilities along the branches.
The probabilities on branches from the same point should sum to 1.
Tree diagrams are useful for independent events and for events with two stages.

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1.What is the sample space of an experiment?
Easy
- AThe set of all possible outcomes of the experiment
- BThe single outcome that actually happens
- CThe list of outcomes that are impossible
- DThe number of times the experiment is repeated
2.A single fair six-sided die is rolled once and the number of pips facing up is recorded. Which set is the sample space?
Easy
- A{1, 2, 3, 4, 5, 6}
- B{0, 1, 2, 3, 4, 5, 6}
- C{2, 4, 6}
- D{1, 2, 3, 4, 5}
3.Two fair coins are tossed. Which set is the sample space, where H = heads and T = tails?
Easy
- A{HH, HT, TH, TT}
- B{HH, TT}
- C{H, T}
- D{HH, HT, TT}
4.Two fair coins are tossed. The event E is that at least one of the coins is heads. Which set lists the outcomes in E?
Medium
- A{HH, HT, TH}
- B{HH}
- C{HT, TH}
- D{HH, HT, TH, TT}
5.A sample space must satisfy certain conditions. Which of the following statements are true? (select all that apply)
Medium
- AThe outcomes must be mutually exclusive
- BThe outcomes must be collectively exhaustive
- CThe sample space must be empty
- DEvery outcome must have the same probability
6.In probability, an event is a subset of the sample space.
Easy
True or false?
7.In a Venn diagram showing a sample space, how are the outcomes of the sample space usually represented?
Medium
- AAs points inside a rectangle
- BAs points outside the rectangle
- CAs arrows pointing to the rectangle
- DAs labels written along the edge of the rectangle
8.In a Venn diagram, the outcomes of an event are shown by which region?
Medium
- AThe points inside the oval representing that event
- BThe points outside the rectangle
- CThe points on the boundary of the rectangle only
- DThe points inside every other oval but not this one

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9.Two fair coins are tossed and the outcome is written with the first coin first, for example HT means the first coin is heads and the second is tails. Which outcome means both coins are tails?
Medium
- ATT
- BTH
- CHT
- DHH
10.A fair six-sided die is rolled once. Which of the following are subsets of the sample space {1, 2, 3, 4, 5, 6} that represent a valid event? (select all that apply)
Medium
- A{2, 4, 6}
- B{1, 3, 5}
- C{0, 1, 2}
- D{7}
- E{1, 2, 3, 4, 5, 6}
11.Match each set notation symbol to its meaning in probability.
Medium
- A ∪ B
- A ∩ B
- A'
- A ∩ B = { }
- the complement of A
- the union of A and B
- A and B are mutually exclusive
- the intersection of A and B
12.Two fair coins are tossed. The event A is 'at least one head' and the event B is 'at least one tail'. Which of the following statements are true? (select all that apply)
Medium
- AA = {HH, HT, TH}
- BB = {HT, TH, TT}
- CA and B are mutually exclusive
- DA ∩ B = {HT, TH}
- EA ∪ B = {HH, HT, TH, TT}
13.Put the steps for drawing a tree diagram for two events in the correct order.
Medium
- Write the probability on each branch
- Draw the first set of branches for the first event
- Multiply along each pair of branches to find each combined outcome
- Draw the second set of branches from each outcome of the first event
14.A fair coin is tossed and a fair six-sided die is rolled. Which of the following are outcomes in the sample space? (select all that apply)
Hard
- AH1
- BT6
- CH7
- DT0
- EH3
15.Match each experiment to its correct sample space.
Medium
- Roll one fair six-sided die
- Toss one fair coin
- Toss two fair coins
- Roll two fair six-sided dice and record the total
- {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
- {1, 2, 3, 4, 5, 6}
- {H, T}
- {HH, HT, TH, TT}
16.Place these events in order of their probability, from smallest to largest, when a single fair six-sided die is rolled.
Medium
- Rolling a 6
- Rolling an even number
- Rolling a number greater than 1
- Rolling any number from 1 to 6
17.A bag contains 3 red counters and 2 blue counters. A counter is taken and replaced, then a second counter is taken. Which of the following statements are true? (select all that apply)
Hard
- AThe two draws are independent events
- BThe probability of red then red is 9/25
- CThe probability of red then blue is 6/25
- DThe probability of at least one blue is 16/25
- EThe two draws are mutually exclusive
18.Match each probability rule to the situation where it is used.
Medium
- Multiplication rule
- Addition rule
- Complement rule
- Mutually exclusive events
- Finding the probability of A or B when A and B cannot both happen
- Finding the probability of A and B for independent events
- Finding the probability that an event does not happen
- Two events that cannot occur at the same time
19.A fair coin is tossed twice. Put these events in order of probability, from smallest to largest.
Medium
- Getting two heads
- Getting exactly one head
- Getting at least one head
- Getting heads or tails on the first toss
20.In a Venn diagram, the rectangle represents the sample space S, circle A represents event A and circle B represents event B. Which of the following statements are true? (select all that apply)
Hard
- AThe region inside both circles represents A ∩ B
- BThe region inside A but outside B represents A ∩ B'
- CThe region outside both circles represents (A ∪ B)'
- DThe whole rectangle represents A ∪ B
- EThe region inside at least one circle represents A ∪ B
21.Match each combined-outcome probability to its value when two fair six-sided dice are rolled.
Medium
- P(total = 2)
- P(total = 7)
- P(total = 12)
- P(total is even)
- 1/36
- 6/36
- 18/36
22.A fair coin is tossed and a fair six-sided die is rolled. How many outcomes are in the sample space?
Easy
- A6
- B12
- C8
- D36
23.A fair coin is tossed and a fair six-sided die is rolled. What is the probability of getting a head and a 4?
Medium
- A\frac{1}{6}
- B\frac{1}{8}
- C\frac{1}{12}
- D\frac{1}{2}
24.In a Venn diagram showing events A and B, which region represents the event 'A and B both occur'?
Medium
- AThe part of A outside B
- BThe part of B outside A
- CThe overlap of A and B
- DThe region outside both A and B
25.Which of the following statements about independent events are true? (select all that apply)
Medium
- AThe outcome of one event does not affect the outcome of the other.
- BThe multiplication rule can be used to find the probability of both events occurring.
- CThe events must be mutually exclusive.
- DThe probability of both events occurring is found by adding their probabilities.
26.In a tree diagram, the probabilities on the branches from any single point must add up to 1.
Easy
True or false?
27.Match each set notation to its description in a Venn diagram.
Medium
- A ∪ B
- A ∩ B
- A'
- The complement of A
- The union of A and B
- The intersection of A and B
28.A bag contains 4 red and 6 blue counters. A counter is taken at random. What is the probability it is red?
Medium
- A\frac{4}{6}
- B\frac{4}{10}
- C\frac{6}{10}
- D\frac{1}{4}
29.A fair six-sided die is rolled twice and the two scores are added. What is the probability that the total score is 8? Give your answer as a fraction.
Medium
- A5/36
- B4/36
- C6/36
- D1/9
30.A fair six-sided die is rolled once. Which of the following events are mutually exclusive with the event 'rolling an even number'? (select all that apply)
Medium
- ARolling a 3
- BRolling a 5
- CRolling a 2
- DRolling an odd number
- ERolling a number greater than 4
31.In a sample space diagram showing the totals when two fair six-sided dice are rolled, the total of 7 is the most likely outcome.
Medium
True or false?
32.Match each probability expression to its value when a fair six-sided die is rolled once.
Medium
- P(even)
- P(not 3)
- P(4 or 5)
- 1/2
- 5/6
- 1/3
33.Place these steps for finding the probability of getting at least one head when two fair coins are tossed in the correct order.
Medium
- List the sample space {HH, HT, TH, TT}
- Identify the outcomes with at least one head: HH, HT, TH
- Count the favourable outcomes: 3
- Divide by the total number of outcomes: 3/4
34.A bag contains 4 red counters and 6 blue counters. A counter is taken at random and not replaced. A second counter is then taken. What is the probability that both counters are red? Give your answer as a fraction.
Hard
- A2/15
- B4/25
- C1/6
- D3/10
35.What is the probability that the total score is 4? Give your answer as a fraction.
Medium
- A1/12
- B1/9
- C1/6
- D1/4
36.A fair six-sided die is rolled once and the number of pips facing up is recorded. How many outcomes are in the sample space?
Easy
- A6
- B2
- C12
- D36
37.For tossing a single coin, the sample space is {H, T}. How many different events can be formed as subsets of this sample space?
Medium
- A4
- B2
- C3
- D1
38.For tossing two coins, the event E = {HH, HT, TH}. Which statement about E is correct?
Medium
- AE is the event that at least one coin is heads
- BE is the event that both coins are heads
- CE is the event that at least one coin is tails
- DE is the event that exactly one coin is heads
39.The outcomes in a sample space must be mutually exclusive, so only one outcome can occur on any single trial.
Easy
True or false?
40.The outcomes in a sample space must be collectively exhaustive, meaning some outcome in the sample space always occurs on every trial.
Easy
True or false?
41.The sample space of an experiment is often labelled with the letter S. Give the other single-letter label that is also commonly used for the sample space.
Easy
42.Complete the sentence about the sample space for rolling a single six-sided die.
Medium
The sample space for rolling one six-sided die is {1, 2, 3, 4, ____, 6}.
43.Two fair six-sided dice are rolled. What is the probability that both dice show a 6? Give your answer as a fraction.
Medium
44.Complete the sentence about the multiplication rule for independent events.
Medium
For two independent events A and B, the probability that both occur is P(A) ____ P(B).
45.In a tree diagram for two events, what name is given to the probability written on the second set of branches, which is calculated given the outcome of the first event?
Medium
46.A fair six-sided die is rolled once. What is the probability that the number of pips facing up is 5? Give your answer as a fraction.
Medium
47.A fair coin is tossed and a fair six-sided die is rolled. What is the probability of getting a tail and a number greater than 4? Give your answer as a fraction.
Medium
48.Two fair six-sided dice are rolled. What is the probability that the total score is 7? Give your answer as a fraction.
Medium
49.A bag contains 5 red counters and 3 blue counters. A counter is taken at random, its colour noted, and it is replaced. A second counter is then taken. What is the probability that both counters are blue? Give your answer as a fraction.
Medium
50.Complete the sentence about combined events.
Medium
For two mutually exclusive events A and B, P(A or B) = P(A) ____ P(B).
51.A fair six-sided die is rolled twice. What is the probability that the total score is 4? Give your answer as a fraction.
Hard
52.In a tree diagram, what name is given to the probability written on the second set of branches, which is calculated given the outcome of the first event?
Medium
53.Two fair coins are tossed. Explain why {HH, HT, TH, TT} is a valid sample space for this experiment.
Hard
54.In a Venn diagram, the rectangle represents the sample space S, and circles A and B represent two events. Describe the region that represents the event 'A and B both occur'.
Medium
55.In a Venn diagram, the rectangle represents the sample space S, and circles A and B represent two events. Describe the region that represents the event 'A and B'.
Hard
56.Describe how a sample space can be represented visually using a rectangle and ovals.
Medium
57.Two fair six-sided dice are rolled. What is the probability that both dice show a 6? Give your answer as a fraction.
Medium
- A-0.9722
- B2.0278
- C0.027777777777777776
- D1.0278
58.A fair six-sided die is rolled once. What is the probability that the number of pips facing up is 5? Give your answer as a fraction.
Medium
- A1.1667
- B2.1667
- C0.1667
- D-0.8333
59.A fair coin is tossed and a fair six-sided die is rolled. What is the probability of getting a tail and a number greater than 4? Give your answer as a fraction.
Medium
- A0.1667
- B2.1667
- C-0.8333
- D1.1667
60.Two fair six-sided dice are rolled. What is the probability that the total score is 7? Give your answer as a fraction.
Medium
- A2.1667
- B-0.8333
- C1.1667
- D0.1667
61.A bag contains 5 red counters and 3 blue counters. A counter is taken at random, its colour noted, and it is replaced. A second counter is then taken. What is the probability that both counters are blue? Give your answer as a fraction.
Medium
- A-0.8594
- B1.1406
- C0.1406
- D2.1406
62.A fair six-sided die is rolled twice. What is the probability that the total score is 4? Give your answer as a fraction.
Hard
- A-0.9167
- B1.0833
- C2.0833
- D0.0833

Ghi chú ôn tập

## Sample Spaces, Venn Diagrams and Tree Diagrams ### Sample space A **sample space** is the set of all possible outcomes of an experiment. It is written using set notation, e.g. S = {H, T} for one coin toss, or S = {1, 2, 3, 4, 5, 6} for one die roll. Outcomes can be numbers, words, letters or symbols. For two coins: S = {HH, HT, TH, TT}. The event "at least one head" is E = {HH, HT, TH}. An **event** is a subset of the sample space. If the outcome is in E, then E has occurred. ### Conditions - Outcomes must be **mutually exclusive** — only one can happen at a time. - Outcomes must be **collectively exhaustive** — something in the sample space always happens. ### Sample space diagrams For two events (e.g. two dice), draw a grid: one event along the top, the other down the side, and fill in every pair. This lists outcomes systematically and shows totals clearly. ### Venn diagrams and set notation - **Union** A ∪ B: in A or B (or both). - **Intersection** A ∩ B: in both A and B. - **Complement** A′: not in A. ### Mutually exclusive events Two events are **mutually exclusive** if they cannot happen together, so A ∩ B = ∅. For these, the **addition rule** applies: P(A or B) = P(A) + P(B) ### Independent events Two events are **independent** if one does not affect the other. For these, the **multiplication rule** applies: P(A and B) = P(A) × P(B) ### Tree diagrams A tree diagram shows two (or more) events in sequence. Each branch is labelled with a probability, and probabilities on branches from the same point add to 1. Multiply along a path to find the probability of that combination; add the results for paths that give the same overall event. **Exam tips** - List outcomes systematically so none are missed. - Check probabilities on each set of branches sum to 1. - Use × along branches, + between separate paths. - Remember: mutually exclusive → add; independent → multiply.

Sample space: The set of all possible outcomes of an experiment, usually written with set notation, e.g. S = {H, T}.
Event: A subset of the sample space. If the outcome is in the event, the event has occurred.
Mutually exclusive outcomes: Outcomes where if one occurs, no other can take place at the same time.
Collectively exhaustive: On every trial, some outcome in the sample space always occurs.
Union (A ∪ B): The set of outcomes in A or B (or both).
Intersection (A ∩ B): The set of outcomes in both A and B.
Complement (A′): The set of outcomes not in A.
Mutually exclusive events: Events that cannot happen together, so A ∩ B = ∅. Their probabilities add: P(A or B) = P(A) + P(B).
Independent events: Events where one does not affect the other. Their probabilities multiply: P(A and B) = P(A) × P(B).
Tree diagram: A diagram showing events in sequence, with probabilities on branches. Multiply along a path, add across paths for the same event.

Thẻ ghi nhớ

What is a sample space?

The set of all possible outcomes of an experiment or random trial. It is usually written using set notation, with each possible outcome listed as an element.

What notation is commonly used to denote a sample space?

S, Ω (omega), or U (for universal set).

What is an event in probability?

A subset of the sample space. If the outcome of the experiment is included in the event, then the event has occurred.

What is the sample space for tossing a single coin?

{H, T}, where H = heads and T = tails.

What is the sample space for tossing two coins?

{HH, HT, TH, TT}, where the order matters (first coin, second coin).

For two coins, what is the event that at least one coin is heads?

{HH, HT, TH}.

What is the sample space for rolling a single six-sided die once?

{1, 2, 3, 4, 5, 6}.

What are the two conditions that the outcomes in a sample space must satisfy?

They must be mutually exclusive (no two can occur together) and collectively exhaustive (one must occur on every trial).

What does it mean for outcomes to be mutually exclusive?

If one outcome occurs, then no other outcome can occur at the same time.

What does it mean for outcomes to be collectively exhaustive?

On every trial, at least one of the outcomes must occur.

How can a sample space be represented visually?

By a rectangle, with outcomes shown as points inside it. Events are shown as ovals enclosing the relevant points.

What is a sample space diagram for two events?

A grid or table that lists all possible combined outcomes of two events, such as rolling two dice.

In a Venn diagram, what does the rectangle represent?

The universal set, which is the sample space of all possible outcomes.

In set notation, what does A ∪ B mean?

The union of A and B: all outcomes that are in A or B (or both).

In set notation, what does A ∩ B mean?

The intersection of A and B: all outcomes that are in both A and B.

In set notation, what does A' mean?

The complement of A: all outcomes in the sample space that are not in A.

What are mutually exclusive events?

Events that cannot happen at the same time. Their intersection is empty, so P(A ∩ B) = 0.

What is the addition rule for mutually exclusive events?

P(A ∪ B) = P(A) + P(B), because the events do not overlap.

What is the general addition rule for any two events?

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

What are independent events?

Events where the occurrence of one does not affect the probability of the other.

What is the multiplication rule for independent events?

P(A and B) = P(A) × P(B).

How do you find the probability of an event from a sample space?

Divide the number of favourable outcomes by the total number of outcomes in the sample space, assuming all outcomes are equally likely.

What is a tree diagram used for?

To show all possible outcomes of two or more events in sequence, with branches for each outcome and probabilities written on the branches.

How do you calculate the probability of a combined outcome from a tree diagram?

Multiply the probabilities along the branches that lead to that outcome.

What is the sum of the probabilities on all branches from a single point in a tree diagram?

1, because the branches represent all possible outcomes at that stage.

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