Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Count samples with a fixed number of selected target items.
- Justify the conclusion "Probability=36/120=3/10" using the stated assumptions.
Before you start
Combinations and equally likely subsets.
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
A box has four red and six blue tokens. Three are selected without replacement. Find the probability of exactly two red.
Why this math matters
Count samples with a fixed number of selected target items. 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:
- Selection is uniform among subsets.
- Tokens are not replaced.
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.
Model sampling without replacement
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A box has four red and six blue tokens. Three are selected without replacement. Find the probability of exactly two red.
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
Total unordered samples=C(10,3)=120
Each three-token subset is equally likely.
Work through the mathematics
Favorable samples=C(4,2)C(6,1)=36
Choose the required red and blue tokens independently as sets.
Check the conclusion
Probability=36/120=3/10
The hypergeometric model accounts for changing composition after each draw.
The result
Probability=36/120=3/10
The hypergeometric model accounts for changing composition after each draw.
Common mistakes to catch
- Do not multiply constant draw probabilities without checking replacement.
- Use matching ordered or unordered counting conventions throughout.
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
What is the probability all three are red?
Show a hint
Choose three from four.
Reveal answer and explanation
1/30
C(4,3)/C(10,3)=4/120.
Practice 2
Why is a binomial model not exact here?
Show a hint
The draws change the box composition.
Reveal answer and explanation
The draws are not independent with constant red probability
Removing a token changes subsequent conditional probabilities.
Take the idea with you
Model an inspection sample from a finite batch.
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: Read probabilities from a generating polynomial
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