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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- State a finite sample space.
- Calculate a union without double-counting.
- Use complements and independence in separate examples.
Before you start
Fractions, set membership, and the meaning of and versus or.
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
For one roll of a fair six-sided die, what is the probability of an even number or a number greater than four?
Why this math matters
A probability calculation starts with a model, not a fraction chosen by appearance. Counting favorable outcomes works when the elementary outcomes are equally likely. When categories overlap, an outcome can qualify in two ways while still representing only one roll. Listing the sample space makes that double-counting visible.
Set up the model
A useful answer starts with clear assumptions:
- The die has outcomes 1 through 6, each with probability 1/6.
- Only one roll is used in the main example.
- Or is inclusive: an outcome satisfying both conditions still qualifies.
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.
When can you count outcomes to find a probability?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For one roll of a fair six-sided die, what is the probability of an even number or a number greater than four?
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.
List the events
A = {2, 4, 6}; B = {5, 6}
Even numbers make up A, while values strictly greater than four make up B. The outcome six belongs to both.
Form the union
A ∪ B = {2, 4, 5, 6}; P(A ∪ B) = 4/6 = 2/3
There are four different favorable outcomes. The fair-die assumption lets each contribute the same one-sixth probability.
Check with the addition rule
P(A) + P(B) − P(A ∩ B) = 3/6 + 2/6 − 1/6 = 4/6
Subtract the overlap because it was counted once in each event total. The excluded outcomes one and three also give a complementary check.
The result
The probability is 2/3 under the stated fair-die model.
If the die were biased, the same outcome lists would remain correct, but their probabilities would need to be added individually. Dividing by six would no longer be justified merely because six faces exist.
Common mistakes to catch
- Adding 3/6 and 2/6 without subtracting the overlap counts six twice.
- Mutually exclusive and independent do not mean the same thing.
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
For one fair-die roll, what is P(even or odd)?
Show a hint
These two events are disjoint and cover the sample space.
Reveal answer and explanation
1
Every possible outcome is either even or odd, so the union is certain.
Practice 2
For two independent fair-coin flips, what is the probability of at least one head?
Show a hint
Use the complement, no heads.
Reveal answer and explanation
3/4
No heads means TT, with probability (1/2)(1/2) = 1/4. Its complement has probability 3/4.
Take the idea with you
List the outcomes, state their probabilities, and identify overlaps before applying a shortcut formula.
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.
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