Set Notation And Probability Diagrams

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レッスンノート

Two-Way Tables

A two-way table compares two characteristics (e.g., year group and language choice).
Add row and column totals, including an overall total in the bottom-right corner.
Probability from the whole table: denominator is the overall total.
Probability from a specific category: denominator is that category total.
Always check that row and column totals sum to the overall total.

Two-way table: language choice by year group

Set Notation

ℰ (or U, ξ) is the universal set – everything under consideration.
n(A) is the number of elements in set A.
A ∩ B is the intersection – elements in both A and B.
A ∪ B is the union – elements in A or B or both.
A' is the complement – elements in ℰ not in A.
Sets can be described using curly brackets: {x : rule}.

Venn Diagrams

A Venn diagram uses a rectangle (ℰ) and circles for sets.
Overlapping circles show the intersection of sets.
Place numbers or elements in each region, starting with the intersection.
Work outwards from the intersection to fill the rest.

Sets A and B

Probabilities from Venn Diagrams

Probability = (number of elements in region) / (total number of elements).
For conditional probability, restrict the denominator to the given set.
Be careful: some given numbers may need to be split between regions.

Spanish and German study (class of 30)

Probability Tree Diagrams

Tree diagrams show outcomes of repeated experiments with two outcomes each.
Probabilities on branches from the same point add to 1.
Multiply along branches to find the probability of a sequence.
Add probabilities of different sequences to find the probability of combined events.
For 'at least one', it is often easier to use 1 − P(none).

Two sets of traffic lights

Combined Probability

P(A and B) = P(A) × P(B) (the 'and rule') for independent events.
P(A or B) = P(A) + P(B) (the 'or rule') for mutually exclusive events.
Events are independent if one does not affect the other.
For 'without replacement', events are not independent – adjust probabilities accordingly.

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練習問題

無料プレビュー — 46問中8問。すべて見るには登録を。

1.ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {1, 4, 9}. What is n(A)?
Easy
- A3
- B4
- C5
- D6
2.Which set notation describes the shaded region in a Venn diagram where only the overlap of A and B is shaded?
Easy
- AA ∩ B
- BA ∪ B
- CA'
- DB'
3.In a Venn diagram, the universal set is represented by a:
Easy
- Arectangle
- Bcircle
- Ctriangle
- Dsquare
4.If P(rain) = 1/3 and P(go fishing | rain) = 3/5, what is P(rain and go fishing)?
Medium
- A1/5
- B3/15
- C2/5
- D1/3
5.In a two-way table, the overall total should be placed in which corner?
Easy
- Abottom-right
- Btop-left
- Cbottom-left
- Dtop-right
6.ℰ = {1,2,3,4,5,6,7,8,9,10,11,12,13,14}. F = {x: x is a factor of 14}. P = {x: x is a prime number less than 14}. What is n(F ∩ P)?
Medium
- A2
- B3
- C4
- D5
7.In a group of 40 students, 24 like football, 19 like cricket, and 10 like football but not cricket. How many students like both football and cricket?
Medium
- A14
- B5
- C10
- D9
8.The probability that Arun plays football is 3/4 and that Bob plays football is 2/5. What is the probability that both play football?
Medium
- A3/10
- B6/20
- C1/2
- D3/5

もっと問題

9.A fair six-sided dice is rolled twice. What is the probability of getting a number greater than 4 both times?
Medium
- A1/9
- B4/36
- C2/6
- D1/3
10.In a class of 30 students, 15 study Spanish, 3 study both Spanish and German, and 7 study neither. How many study only German?
Hard
- A8
- B5
- C12
- D10
11.ℰ = {x: x is a positive integer less than 20}. A = {x: x is an even number}. B = {x: x is a multiple of 3}. What is n(A ∪ B)?
Medium
- A13
- B10
- C15
- D12
12.Ali rolls a fair six-sided dice twice. What is the probability that he gets a number greater than 4 at least once?
Hard
- A5/9
- B4/9
- C1/3
- D2/3
13.Bag A has 0.6 probability of green, bag B has 0.2 probability of green. One bead taken from each bag. What is the probability of getting one of each colour?
Medium
- A0.56
- B0.44
- C0.24
- D0.12
14.ℰ = {positive integers less than 16}. X = {prime numbers}. Y = {multiples of 3}. What is n(Y')?
Medium
- A11
- B10
- C12
- D9
15.ℰ = {natural numbers ≤ 15}. F = {factors of 12}. O = {odd numbers}. A number is chosen at random from ℰ. What is the probability it is in O?
Medium
- A8/15
- B7/15
- C1/2
- D5/15
16.A survey of 60 children is done. 24 like football, 19 like cricket, and 10 like football but not cricket. How many like both football and cricket?
Medium
- A14
- B9
- C5
- D10
17.The Venn diagram shows frequencies. Region A only: 10, A∩B: 5, B only: 8, neither: 2. What is P(A ∪ B)?
Medium
- A23/25
- B15/25
- C13/25
- D18/25
18.A bag contains 3 blue and 8 red counters. A counter is taken, replaced, then another taken. What is the probability of getting one of each colour?
Hard
- A48/121
- B24/121
- C64/121
- D30/121
19.In a two-way table, what does the number in the bottom-right corner represent?
Easy
- AThe total number of items
- BThe sum of the first row
- CThe sum of the first column
- DThe number of items in the last category
20.ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {1, 2, 3, 4}. B = {3, 4, 5, 6}. What is n(A ∩ B)?
Easy
- A2
- B4
- C8
- D6
21.Which of the following is the correct set notation for the complement of set A?
Easy
- AA'
- BA ∩ B
- CA ∪ B
- Dn(A)
22.The two-way table shows the number of students studying Spanish or German in Year 12 and Year 13. What is the probability that a randomly selected student studies Spanish?
Easy
- A20/55
- B15/55
- C20/30
- D15/30
23.A bag contains 3 blue and 8 red counters. A counter is taken at random and replaced. Then another counter is taken. What is the probability that both counters are red?
Medium
- A64/121
- B24/121
- C8/11
- D16/121
24.ℰ = {1,2,3,4,5,6,7,8,9,10,11,12,13,14}. F = {x: x is a factor of 14}. P = {x: x is a prime number less than 14}. What is n(F ∪ P)?
Medium
- A10
- B9
- C11
- D8
25.Which symbol is used to represent the universal set?
Easy
- Aℰ
- BU
- Cξ
- DAll of the above
26.What does n(A) represent?
Easy
- AThe number of elements in set A
- BThe complement of set A
- CThe union of A and something
- DThe intersection of A and something
27.In a Venn diagram, which region represents A ∩ B?
Easy
- AThe overlap of A and B
- BEverything in A
- CEverything in B
- DEverything outside A and B
28.A tree diagram shows the probability of rain is 1/3. If it rains, the probability of going fishing is 3/5. If it does not rain, the probability of going fishing is 3/4. What is the probability that it rains and Amira goes fishing?
Medium
- A1/5
- B1/3
- C3/5
- D1/2
29.The complement of set A, written A', contains all elements in the universal set that are not in A.
Easy
True or false?
30.In a two-way table, the number of students who study Spanish is 20 out of 55 total. What is the probability that a randomly selected student studies Spanish?
Easy
- A20/55
- B20/35
- C35/55
- D20/20
31.If P(A) = 0.6 and P(B) = 0.5, then P(A and B) must be 0.3.
Medium
True or false?
32.Match each set notation to its correct description.
Medium
- A ∩ B
- A ∪ B
- A'
- n(A)
- Number of elements in set A
- Elements in A or B or both
- Elements not in A
- Elements in both A and B
33.Arrange the following steps in the correct order to find the probability that a student chosen at random from a class studies Spanish, given that they are in Year 13, using a two-way table.
Medium
- Identify the total number of students in Year 13.
- Identify the number of Year 13 students who study Spanish.
- Divide the number of Year 13 Spanish students by the total Year 13 students.
- Read the two-way table to find the relevant frequencies.
34.In a Venn diagram showing sets A and B, which of the following regions are shaded to represent A ∪ B? (Select all that apply)
Medium
- ARegion of A only
- BRegion of B only
- CRegion of A ∩ B
- DRegion outside both A and B
35.In a probability tree diagram, the probabilities on branches that split from the same point must sum to 1.
Easy
True or false?
36.The two-way table shows the number of students in Year 12 and Year 13 who study Spanish or German. A student is chosen at random from Year 12. What is the probability that they study German? | | Spanish | German | Total | |-------|---------|--------|-------| |Year 12| 15 | 10 | 25 | |Year 13| 5 | 25 | 30 | |Total | 20 | 35 | 55 |
Medium
- A10/55
- B10/25
- C10/35
- D25/55
37.The Venn diagram shows the number of elements in each region. What is P(A ∩ B)? A: 10 only, A∩B: 5, B: 8 only, Neither: 2
Medium
- A5/25
- B5/23
- C5/15
- D5/13
38.A coin is flipped 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
- C1/4
- D1/12
39.Write the set of positive integers less than 5 using curly bracket notation.
Easy
40.ℰ = {1,2,3,4,5,6,7,8,9,10}. A = {1,2,3,4}. B = {3,4,5,6}. What is n(A ∪ B)?
Medium
41.A fair six-sided dice is rolled twice. What is the probability of getting a 5 both times? Give your answer as a fraction in simplest form.
Easy
42.A bag contains 5 red and 3 blue counters. Two counters are taken at random without replacement. What is the probability that both counters are the same colour? Give your answer as a fraction in simplest form.
Hard
43.Complete the sentences using set notation.
Easy
The set of elements that are in set A or set B or both is called the ____ of A and B. The set of elements that are in both set A and set B is called the ____ of A and B.
44.ℰ = {1,2,3,4,5,6,7,8,9,10}. A = {1,2,3,4}. B = {3,4,5,6}. What is n(A ∪ B)?
Medium
- A8
- B5
- C7
- D6
45.A fair six-sided dice is rolled twice. What is the probability of getting a 5 both times? Give your answer as a fraction in simplest form.
Medium
- A1/36
- B1/6
- C1/12
- D1/9
46.A bag contains 5 red and 3 blue counters. Two counters are taken at random without replacement. What is the probability that both counters are the same colour? Give your answer as a fraction in simplest form.
Hard
- A3/28
- B13/28
- C1/2
- D15/28

復習ノート

## Set Notation & Probability Diagrams – Quick Revision ### Key Notation - **ℰ** = universal set (everything) - **n(A)** = number of elements in set A - **A ∩ B** = intersection (elements in both A and B) - **A ∪ B** = union (elements in A or B or both) - **A'** = complement (elements not in A) ### Venn Diagrams - Rectangle = ℰ, circles = sets. - Fill numbers carefully: given totals may need splitting (e.g., 10 cat owners, 6 also have dog → 4 cat only). - Probabilities: count elements in region / total elements. - Conditional: P(in B | in A) = n(A∩B) / n(A). ### Two-Way Tables - Add row and column totals, overall total in bottom right. - Probability = (number in category) / (total or row/column total if restricted). ### Tree Diagrams - Each branch: probability, P(not) = 1 – P(event). - Multiply along branches for "and". - Add between separate cases for "or". - For "at least one": 1 – P(none). ### Combined Probability - **Independent events**: P(A and B) = P(A) × P(B). - P(A or B) = P(A) + P(B) for mutually exclusive. - Rephrase with "and"/"or". - Watch out for "without replacement" – not independent. ### Quick Tips - Check totals in tables/venn diagrams. - Don't cancel fractions when multiplying along tree branches – keep same denominator for easy addition. - For "given that", denominator is the condition's total. --- ## Key Terms

Universal set (ℰ): The set of all elements under consideration.
n(A): The number of elements in set A.
Intersection (A ∩ B): The set of elements that are in both A and B.
Union (A ∪ B): The set of elements that are in A or B or both.
Complement (A'): The set of all elements in ℰ that are not in A.
Venn diagram: A diagram using circles inside a rectangle to represent sets and their relationships.
Two-way table: A table that displays frequencies for two categorical variables.
Tree diagram: A diagram with branches showing probabilities of sequential events.
Independent events: Events where the outcome of one does not affect the other.
Combined probability: Probability of successive events, using multiplication for 'and' and addition for 'or'.

フラッシュカード

What is the universal set symbol?

ℰ (or ξ, U). It contains all elements under consideration.

What does n(A) represent?

The number of elements in set A.

What does A ∩ B mean?

The intersection of A and B: elements in both A and B.

What does A ∪ B mean?

The union of A and B: elements in A or B or both.

What does A' (complement) represent?

All elements in the universal set ℰ that are not in A.

In a Venn diagram, which region represents A ∩ B?

The overlapping region of the circles for A and B.

In a Venn diagram, which region represents A ∪ B?

All regions inside at least one of the circles A or B.

How do you find probability from a Venn diagram showing frequencies?

Add the frequencies in the desired region(s) and divide by the total frequency.

How do you find probability from a Venn diagram showing individual elements?

Count the number of elements in the desired region(s) and divide by the total number of elements.

In a two-way table, how do you find the probability a student is in a specific category?

Divide the number in that category by the overall total (or by the row/column total if conditional).

What is the 'and' rule for independent events?

P(A and B) = P(A) × P(B).

What is the 'or' rule for mutually exclusive events?

P(A or B) = P(A) + P(B).

On a tree diagram, how do you find the probability of a sequence of outcomes?

Multiply the probabilities along the branches from left to right.

On a tree diagram, how do you find the probability of either one sequence or another?

Add the probabilities of the separate sequences.

How do you find P(at least one) using complements?

P(at least one) = 1 - P(none).

What does it mean if events are independent?

The outcome of one event does not affect the probability of the other.

What is the probability that a fair dice lands on a number greater than 4?

2/6 = 1/3 (numbers 5 and 6).

In a Venn diagram with frequencies, how do you find P(A given B)?

Divide the frequency in A ∩ B by the total frequency in B.

Write the set of even numbers less than 10 using set notation.

{x : x is an even number, x < 10} or {2, 4, 6, 8}.

What is the complement of a set?

All elements in the universal set that are not in the set.

How do you complete a two-way table?

Fill in given numbers, then use row and column totals to find missing values; ensure all totals match.

In a tree diagram, what do the probabilities on each pair of branches add to?

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