PROBABILITY · TEXTBOOK LESSON
Probability measures uncertainty on a scale from 0 to 1. A probability model lists possible outcomes and assigns probabilities consistently so that the total probability is 1.
Learning objectives
By the end of this lesson, you should be able to:
- Express probabilities as fractions, decimals or percentages.
- Find probabilities from equally likely outcomes.
- Use complements, sample spaces, tree diagrams or set notation where appropriate.
- Distinguish independent and dependent events.
1. Key vocabulary and definition
Definition
For equally likely outcomes, P(event) = number of favourable outcomes / total number of possible outcomes. Probabilities satisfy 0 ≤ P(A) ≤ 1.
outcomeeventsample spacecomplementindependentconditional
2. Essential facts and rules
These are the facts you should know before attempting the worked example.
Key mathematical facts
- P(not A) = 1 - P(A).
- Mutually exclusive events cannot occur together.
- For independent events, P(A and B)=P(A)P(B).
3. Standard method
A reliable method helps prevent errors and makes your reasoning easy to follow.
1
Define the event precisely.
2
List or represent the complete sample space.
3
Identify favourable outcomes.
4
Apply the appropriate probability rule.
5
Check that probabilities are between 0 and 1 and totals are consistent.
4. Worked example
Worked example
A fair die is rolled. Find the probability of obtaining a number greater than 4.
1
Sample space: {1,2,3,4,5,6}.
2
Favourable outcomes greater than 4 are {5,6}.
3
There are 2 favourable outcomes out of 6.
4
P(number > 4)=2/6=1/3.
5. Common mistakes
Watch out for these
- Starting calculations before deciding what the question is asking.
- Skipping important steps or changing notation midway through a solution.
- Accepting an answer without checking whether its size, sign or unit is sensible.
6. Video lesson
This topic is ready for a matching video lesson. The written textbook lesson is fully available now; additional videos can be added to the same lesson template.
7. Practice exercise
Complete these without looking at the answers first.
- A fair coin is tossed. Find P(head).
- A fair die is rolled. Find P(not 6).
- A bag has 3 red and 7 blue counters. Find P(red).
- If P(A)=0.35, find P(not A).
- Two fair coins are tossed. Find P(two heads).
Show answers
- 1/2
- 5/6
- 3/10
- 0.65
- 1/4
8. Chapter summary
- Understand the underlying idea before memorising a procedure.
- Use precise mathematical notation and show a logical method.
- Always check the final result.