Learn with Amar
Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Test the difference between two-way and three-way independence.
- Justify the conclusion "The three are not mutually independent" using the stated assumptions.
Before you start
Fair coin outcomes and conditional probability.
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
Let X,Y be independent fair bits and Z=0 when X=Y, Z=1 otherwise. Are X,Y,Z mutually independent?
Why this math matters
Test the difference between two-way and three-way 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:
- X and Y are independent fair bits.
- Z records their exclusive-or relation.
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.
Find variables that are pairwise independent but not jointly independent
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Let X,Y be independent fair bits and Z=0 when X=Y, Z=1 otherwise. Are X,Y,Z mutually independent?
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
The four triples are (0,0,0),(0,1,1),(1,0,1),(1,1,0), each with probability 1/4
Listing the equally likely bit pairs determines Z.
Work through the mathematics
Every pair has all four value combinations equally likely
Each pair is independent with fair-bit marginals.
Check the conclusion
The three are not mutually independent
P(X=0,Y=0,Z=0)=1/4 differs from the product 1/8, and knowing X,Y determines Z.
The result
The three are not mutually independent
P(X=0,Y=0,Z=0)=1/4 differs from the product 1/8, and knowing X,Y determines Z.
Common mistakes to catch
- Pairwise independence does not imply mutual independence.
- Checking only marginal probabilities is insufficient.
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 P(Z=1)?
Show a hint
Count unequal input pairs.
Reveal answer and explanation
1/2
Two of the four equally likely pairs differ.
Practice 2
What is P(Z=1|X=1,Y=0)?
Show a hint
Z is determined by the two bits.
Reveal answer and explanation
One
The stated inputs are unequal.
Take the idea with you
Audit a multi-sensor independence assumption by testing joint patterns, not only pairs.
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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