Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use inclusion–exclusion without assuming independence.
- Justify the conclusion "P(A∪B)=0.8; P(neither)=0.2" using the stated assumptions.
Before you start
Probability axioms and sets.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
If P(A)=0.6, P(B)=0.5, and P(A∩B)=0.3, find P(A∪B).
Why this math matters
Use inclusion–exclusion without assuming independence. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- All probabilities refer to the same probability space.
- The intersection probability is supplied rather than inferred.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Correct double counting in event unions
PausedQuestion: Start with the question. Paused.
Question
Start with the question
If P(A)=0.6, P(B)=0.5, and P(A∩B)=0.3, find P(A∪B).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
P(A)+P(B)=1.1
Outcomes belonging to both events have been counted twice.
Work through the mathematics
P(A∪B)=0.6+0.5−0.3
Subtract the shared intersection once.
Check the conclusion
P(A∪B)=0.8; P(neither)=0.2
The complement has the remaining probability, and the union stays within the probability bounds.
The result
P(A∪B)=0.8; P(neither)=0.2
The complement has the remaining probability, and the union stays within the probability bounds.
Common mistakes to catch
- Independence and disjointness are different concepts.
- Probabilities above one indicate inconsistent assumptions.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Are A and B independent here?
Show a hint
Compare the intersection with the product.
Reveal answer and explanation
Yes
0.6·0.5=0.3 equals the given intersection.
Practice 2
If A and B were disjoint with these marginal probabilities, would the data be valid?
Show a hint
Add the disjoint probabilities.
Reveal answer and explanation
No
The union would have probability 1.1, which is impossible.
Take the idea with you
Audit a survey summary whose overlapping categories add to more than 100 percent.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Update odds with a likelihood ratio
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