CIE IGCSE Mathematics (0580) Conditional probability Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Conditional probability Study Notes
LEARNING OBJECTIVE
- Conditional Probability
Key Concepts:
- Conditional Probability
Conditional Probability
Conditional Probability
Conditional probability is the probability of an event occurring, given that another event has already happened. It is especially useful when outcomes are not equally likely due to some condition or prior knowledge.
Important:
- You do not need to use notation like \( P(A|B) \).
- Use given data and context to calculate probabilities from Venn diagrams, tree diagrams, or tables.
1. Using Venn Diagrams
A Venn diagram visually shows the relationship between different sets. To calculate conditional probability, focus only on the relevant part (i.e. inside the circle or section specified).
Example:
A class of 40 students was surveyed. 25 students like Maths, 18 like Science, and 10 like both. A student is selected at random. Given that the student likes Maths, what is the probability that the student also likes Science?
▶️ Answer/Explanation
Total who like Maths = 25
Like both = 10
Required probability = \( \dfrac{10}{25} = \boxed{\dfrac{2}{5}} \)
Example:
Out of 100 students, 60 play football, 30 play basketball, and 20 play both. What is the probability that a student plays basketball given that they play football?
▶️ Answer/Explanation
Football = 60
Both = 20
Required probability = \( \dfrac{20}{60} = \boxed{\dfrac{1}{3}} \)
2. Using Tree Diagrams
Tree diagrams are used to list outcomes in stages. For conditional probability, restrict attention to paths consistent with the given condition.
Example:
A bag contains 4 red and 6 blue balls. Two are drawn one after another without replacement. What is the probability that the second is red given the first was blue?
▶️ Answer/Explanation
If first is blue (6/10), remaining red = 4, total = 9
\( P = \dfrac{4}{9} \)
Example:
A coin is tossed twice. If the first toss is a head, what is the probability that the second toss is also a head?
▶️ Answer/Explanation
P = \( \dfrac{1}{2} \)
Reason: coin tosses are independent events.
3. Using Tables
Tables can be used to organise data. To find conditional probability, use the relevant row or column total as your denominator.
Example:
The table shows results of 40 students:
| Pass | Fail | Total | |
|---|---|---|---|
| Maths | 18 | 7 | 25 |
| English | 10 | 5 | 15 |
| Total | 28 | 12 | 40 |
If a student is known to have passed, find the probability they are from the Maths group.
▶️ Answer/Explanation
Passed total = 28, Maths passed = 18
\( \dfrac{18}{28} = \boxed{\dfrac{9}{14}} \)
Example:
Survey data:
| Own a Pet | No Pet | Total | |
|---|---|---|---|
| Boys | 12 | 8 | 20 |
| Girls | 16 | 4 | 20 |
If a student is selected and known to be a pet owner, what is the probability the student is a girl?
▶️ Answer/Explanation
Total pet owners = 12 + 16 = 28
Girls with pets = 16
Probability = \( \dfrac{16}{28} = \boxed{\dfrac{4}{7}} \)