Conditional probability definition
In a class of 30 students, 15 study maths and 10 study physics. 6 study both. Find P(physics | maths).
Given someone studies maths, the probability they also study physics is 6 out of 15 = .
Year 11
Conditional probability and linked-event revision
Calculate accurately, explain dependencies clearly and solve unfamiliar probability contexts.
Lesson overview
Statistics — probability and events
Conditional probability is the probability of an event given that another event has already occurred, written P(A|B) = P(A and B) / P(B). Tree diagrams and two-way tables are essential tools for visualising linked events. Without replacement problems (e.g., drawing cards from a deck without returning them) change probabilities as the sample space shrinks. Key skill: identifying whether events are independent (probability unchanged) or dependent (probability changes after first event). Always check: Does the second outcome depend on the first? If yes, use conditional probability formulas or systematic tree diagrams.
Tree diagram for dependent events (without replacement)
A bag has 5 red and 3 blue balls. Draw two without replacement. Find P(both red).
Without replacement: after the first draw, the second probability changes because the sample size decreased.
Two-way table for linked events
Survey: 40 men, 30 women. 20 men and 15 women have a gym membership. Find P(member | male).
| Member | Not member | Total | |
|---|---|---|---|
| Male | 20 | 20 | 40 |
| Female | 15 | 15 | 30 |
| Total | 35 | 35 | 70 |
From the 'Male' row: 20 out of 40 males are members.
Independent vs dependent events
Event A: rolling a 6 on a die. Event B: coin lands heads. Are they independent?
Independent events: P(A and B) = P(A) × P(B). If the equation holds, they're independent.
Worked example — conditional probability in context
A school has 100 Year 11 students. 60 play sports, 45 do music, 30 do both. A student is chosen at random. (a) Find P(music | sports). (b) Are 'plays sports' and 'does music' independent events?
- Part (a): Use the conditional probability formula: P(Music | Sports) = P(Both) / P(Sports).
- P(Both) = = 0.3 (students doing both).
- P(Sports) = = 0.6 (all students playing sports).
- P(Music | Sports) = 0.3 / 0.6 = 0.5 or .
- Part (b): Check if P(Sports) × P(Music) = P(Both).
- P(Sports) × P(Music) = 0.6 × 0.45 = 0.27.
- P(Both) = 0.3, which is not equal to 0.27.
- Conclusion: The events are dependent (not independent).
Try it
Use the examples carefully, then choose the answer.
Question 1
A bag has 3 red and 2 blue balls. Draw one, don't replace it, draw again. P(1st red, 2nd blue)?
💡 First draw: 3 red out of 5. Second draw: 2 blue out of 4 remaining.
Question 2
P(A) = 0.4, P(B) = 0.3. If A and B are independent, what is P(A and B)?
💡 Independent: P(A and B) = P(A) × P(B) = 0.4 × 0.3 = 0.12.
Question 3
From a two-way table: 20 adults own a car, 30 adults in total. P(car owner | adult)?
💡 Given adult, how many own a car? 20 out of 30.
Question 4
A deck has 52 cards. Draw two without replacement. P(2 aces)?
💡 First ace: 4 out of 52. Second ace (given first was drawn): 3 out of 51.
Question 5
Events X and Y where P(X) = 0.2, P(Y) = 0.5, P(X and Y) = 0.1. Are they independent?
💡 Check: 0.2 × 0.5 = 0.1. But 0.1 ≠ 0.1? Actually this IS true, so they ARE independent.
Common mistakes
Watch for these when working through the lesson.
- Forgetting to adjust the sample size after removing items (without replacement).
- Confusing P(A and B) with P(A | B); conditional probability requires dividing by P(B).
- Assuming all events are independent when they might be dependent; always check the context.
- Not identifying when to use a tree diagram vs a two-way table; both work, but one may be clearer.
Related topics
These ideas fit closely with this lesson.
- Mutually exclusive and independent events
- Sample space and outcomes
- Probability rules and formulas
- Venn diagrams and counting