Conditional Probability
Recognize what the condition changes, choose the correct denominator, and solve table, tree, replacement, independence, and screening problems.
Conditional Probability
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Treat the condition as a new universe
In P(A | B), event B is already known to have occurred. Only outcomes inside B remain possible.
P(A | B) = P(A ∩ B)P(B)The overlap stays in the numerator. The probability of the given event becomes the denominator.
Condition
The event after the vertical bar; it restricts the sample space.
Intersection
The outcomes satisfying both events.
Without replacement
Probabilities change after an object is removed.
With replacement
The original probabilities return before the next draw.
Independence
P(A|B) = P(A), equivalently P(A∩B) = P(A)P(B).
Read the words after “given that” first
1. Circle the condition. It determines the new denominator.
2. Count outcomes satisfying both conditions. That is the numerator.
3. For sequential draws, decide whether the first object is replaced.
4. For “at least one,” try the complement.
5. Do not reverse P(A|B) and P(B|A).
Why it works
The arithmetic is often short. Most errors come from keeping the original total even after the question restricts the group.
Five forms you should recognize
Given Grade 11, use 60 as the denominator. Given Science club, use 40.
Each path probability is found by multiplying; different qualifying paths are then added.
For P(A|B), compare the overlap with all of B—not with the entire rectangle.
Check before you commit
- Using the original total after a condition has narrowed the group
- Reversing P(A|B) and P(B|A)
- Forgetting to change the denominator without replacement
- Multiplying mutually exclusive alternatives instead of adding them
- Assuming events are independent merely because they are different
- Ignoring false positives when interpreting a positive test
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Conditional Probability FAQ
What does the vertical bar mean?
It means “given that.” Everything after the bar is treated as known.
Are independent and mutually exclusive the same?
No. Independent events do not affect one another; mutually exclusive events cannot occur together.
When should I use a tree diagram?
Use one for sequential events, especially when probabilities change between stages.
Continue your mathematics review.
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