Learning Objectives
  • Use set notation and describe sets accurately.
  • Represent sets using Venn diagrams.
  • Calculate unions, intersections and complements.
  • Use the probability scale and calculate probabilities of single events.
  • Use relative frequency and expected frequency as introductory probability tools.
Key Terms
Universal set
The complete set of objects currently under consideration, written \xi or U.
Subset
A set whose elements all belong to another set.
Union
Elements in A or B or both, written A\cup B.
Intersection
Elements common to both sets, written A\cap B.
Complement
Elements in the universal set but not in a specified set, written A\prime.
Probability
A numerical measure from 0 to 1 of how likely an event is.
Relative frequency
Observed event frequency divided by number of trials.
8th Edition Chapter Map
  • Set descriptions and notation
  • Venn diagrams
  • Union, intersection and complement
  • Single-event probability
  • Relative and expected frequency
Describing Sets

A set is a collection of distinct objects called elements. Use braces for roster notation, for example A=\{2,4,6,8\}. Repetition does not create a new element. The empty set is \varnothing. The number of elements in A is n(A).

Notation Meaning
x\in A x is an element of A
x\notin A x is not an element of A
A\subseteq B A is a subset of B
A\cup B A union B
A\cap B A intersection B
A\prime Complement of A
Venn Diagrams

The rectangle represents the universal set and circles represent subsets. When filling a two-set diagram from totals, place the intersection first to avoid double counting, then the exclusive regions, then the outside region.

Worked Example: Two Sets

Question: In a class of 40 students, 23 study French, 18 study German and 9 study both. Find how many study neither.

  1. French only =23-9=14.
  2. German only =18-9=9.
  3. Union =14+9+9=32.
  4. Neither =40-32=8.

Answer: 8 students.

Inclusion-Exclusion
n(A\cup B)=n(A)+n(B)-n(A\cap B)

The intersection is subtracted once because it was counted in both n(A) and n(B).

The Probability Scale

Probability 0 means impossible and probability 1 means certain. The probability of an event not occurring is the complement of its probability.

P(A\prime)=1-P(A)
Worked Example: Single Event

Question: A bag contains 5 red, 3 blue and 2 green counters. One counter is selected at random. Find the probability it is blue and the probability it is not blue.

  1. Total counters =10.
  2. P(\text{blue})=3/10.
  3. P(\text{not blue})=1-3/10=7/10.

Answer: 3/10 and 7/10.

Probabilities From Tables And Venn Diagrams

Probability is favourable outcomes divided by total equally likely outcomes. A Venn diagram can provide the counts. Read carefully whether “A or B” includes the intersection; in probability, A\cup B normally includes both.

Relative Frequency

Experimental probability is estimated by relative frequency. As the number of trials grows, it often becomes more stable, but it need not equal the theoretical probability exactly.

\text{relative frequency}=\frac{\text{frequency of event}}{\text{number of trials}}
Worked Example: Estimate Probability

Question: A spinner lands on blue 84 times in 240 spins. Estimate P(\text{blue}).

  1. 84/240=0.35.

Answer: The estimated probability is 0.35.

Expected Frequency

Multiply probability by the number of trials. The answer is an expected long-run value and may not be a whole number for a single experiment, though context may require an estimate.

\text{expected frequency}=\text{probability}\times\text{number of trials}
Worked Example: Expected Number

Question: The probability a manufactured item is defective is 0.018. Estimate the number defective in 2 500 items.

  1. 0.018\times2500=45.

Answer: About 45 items.

Fair, Biased And Random

A fair device gives intended outcomes equal chances. A biased device does not. Random selection means each eligible outcome is selected without a predictable or deliberate preference.

Scope Note

This chapter introduces set notation and single-event probability. Combined-event probability using tree diagrams and more advanced set problems appears in the 8th Edition D4 volume.

Examination Guidance
  • Fill intersections before exclusive regions in Venn diagrams.
  • Check that all Venn regions add to the universal total.
  • Give probability as a fraction, decimal or percentage between 0 and 1.
  • Use the complement rule when “not” is easier than direct counting.
Common Mistakes
  • Adding n(A) and n(B) without subtracting the overlap.
  • Putting repeated elements into a set more than once.
  • Giving a probability greater than 1.
  • Treating relative frequency as an exact theoretical probability.
Knowledge Check And Practice

1. Let A=\{1,2,3,4\} and B=\{3,4,5\}. Find A\cap B.

Answer: \{3,4\}.

2. For the same sets, find A\cup B.

Answer: \{1,2,3,4,5\}.

3. If P(A)=0.37, find P(A\prime).

Answer: 0.63.

4. A die is fair. Find the probability of an even number.

Answer: 3/6=1/2.

5. An event occurs 72 times in 300 trials. Estimate its probability.

Answer: 72/300=0.24.
Extended Worked Practice
Two-Set Counting

Question: In a group of 50 students, 31 study French, 24 study German and 12 study both. Find the number who study at least one language and the number who study neither.

  1. n(F\cup G)=31+24-12=43.
  2. Neither =50-43=7.

Answer: 43 study at least one; 7 study neither.

Complement Probability

Question: The probability that a machine produces a defective item is 0.028. Find the probability that an item is not defective and the expected number of non-defective items in 2500.

  1. P(\text{not defective})=1-0.028=0.972.
  2. Expected number =2500(0.972)=2430.

Answer: 0.972 and 2430.

Estimating Probability From Trials

Question: A spinner lands on blue 186 times in 600 spins. Estimate P(\text{blue}) and predict the number of blues in 1500 further spins.

  1. Relative frequency =186/600=0.31.
  2. Prediction =1500(0.31)=465.

Answer: Estimated probability 0.31; predicted frequency 465.

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