LGT5101 Chap.4 Event Probability and Conditional Reasoning
Event Probability and Conditional Reasoning
Translate words into event regions
Probability begins with an experiment, a complete sample space and named events. Basic outcomes must be mutually exclusive, carry probabilities between zero and one and total one.
A model may use equally likely theoretical outcomes, observed relative frequencies or informed judgement; label the source because these probabilities do not carry identical evidential support.
Match language to set operations
The union rule answers A or B and corrects double counting by subtracting the overlap.
A complement replaces a difficult event with its opposite, making at least one equal to one minus none. An intersection represents both events.
Draw or tabulate regions before substituting numbers, and verify that a union is no smaller than either component while an intersection is no larger than either component.
Change the denominator when information arrives
Conditional probability restricts the sample space to the given event. P(A given B) therefore uses the overlap in its numerator and P(B) in its denominator.
Independence means conditioning does not change a probability and permits the simple product P(A)P(B) for an intersection. It is different from mutual exclusivity: two non-empty mutually exclusive events are dependent because observing one makes the other impossible.
An observed conditional difference shows association, not necessarily causation.
Count outcomes in defensible stages
Permutations count ordered arrangements, while combinations count selected groups. Test the choice by reversing two items: if the reversed result is genuinely different, order matters. Multi-stage problems can combine an unordered selection with an ordered role, so separate the stages and multiply.
Probability trees use conditional branch probabilities, multiply along a path and add across disjoint paths. Naming the reference group, checking branch totals and interpreting the final event in words prevent most formula reversals.
What this chapter covers
- 01
Experiments and sample spaces
- 02
Events and complements
- 03
Permutations and combinations
- 04
Union and intersection
- 05
Conditional probability
- 06
Product rule
- 07
Mutual exclusivity
- 08
Independence
Worked example · free
Urgent orders and late arrivals
- 3Calculate the union without double counting.
- 3Calculate lateness given urgency.
- 3Compare conditional and marginal probabilities.
Key terms
- Sample Space
- The set of all basic outcomes of an experiment.
- Union
- The event that at least one of the named events occurs.
- Intersection
- The event that all named events occur together.
- Conditional Probability
- An event probability calculated inside a restricted sample space.
- Mutual Exclusivity
- A relationship in which two events cannot occur together.
Event Probability and Conditional Reasoning FAQ
Why is overlap subtracted in the union rule?
Adding P(A) and P(B) counts outcomes in their intersection twice, once inside each event. Subtracting P(A intersection B) once leaves every outcome in the union counted exactly once. If the events are mutually exclusive, the overlap is zero.
Are mutually exclusive events independent?
Two non-empty mutually exclusive events are not independent. Once one occurs, the other becomes impossible, so its conditional probability changes to zero. Independence allows both events to occur together and requires that learning one does not change the other's probability.
How do I choose permutation or combination?
Ask whether reversing two selected objects creates a different outcome. If AB and BA represent different assigned positions or sequences, order matters and a permutation applies. If both represent the same selected group, order does not matter and a combination applies.
Exam move
Translate ten probability phrases into event notation: and, or, given, neither, at least one, exactly one and more than. Draw overlapping regions for each before calculating. For counting questions, write one reversed pair to test whether order matters. For independence, verify either a conditional equality or the product equality and state which one you used.
Build a two-by-two table from one union problem and reconcile all four cells to one. Read both P(A given B) and P(B given A) aloud with their reference groups to expose a reversed denominator. Construct a two-stage tree whose second probabilities differ across first branches, multiply along paths and add only disjoint paths. Then make the branches equal and show how independence changes the calculation.
Finish each result with three checks: the probability lies in bounds, an intersection does not exceed either component and a union is at least as large as each component. State association separately from any causal hypothesis. Recalculate one result from both the table and the tree, and investigate any disagreement before accepting either route.
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