Conditional Probability: Shrink the World First
What changes when new information restricts the world in which probability is measured?
How this lesson unfolds.
Begin with What exactly is random? The sequence then connects One outcome is an ordered pair and builds toward Break complex notation into small worlds.
What exactly is random?
Draw two balls without replacement from 10 red and 10 blue distinguishable balls; order matters.
- Learning goalExplain experiment using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionOne outcome is an ordered pair
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The visual connects exact notation to nested sets, counts, or a probability tree.
What belongs in the sample space?
An outcome records both draws in order, such as (R1,B4), so the full sample space contains 20x19=380 outcomes.
An outcome records both draws in order, such as (R1,B4), so the full sample space contains 20x19=380 outcomes.
- Learning goalExplain ordered outcomes using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionAn event contains many outcomes
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The visual connects exact notation to nested sets, counts, or a probability tree.
Why is an event not one result?
A1 is the set of all outcomes whose first ball is red; A2 is the set whose second ball is red.
- Learning goalExplain events as sets using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionThe comma means AND
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The visual connects exact notation to nested sets, counts, or a probability tree.
What does P(A1,A2) mean?
The comma is shorthand for intersection: both the first and second balls are red.
- Learning goalExplain intersection notation using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionThe conditioning bar restricts the world
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The visual connects exact notation to nested sets, counts, or a probability tree.
What does P(A2 | A1) ask?
First enter the world where A1 is true, then measure the part of that smaller world also inside A2.
- Learning goalExplain conditioning meaning using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionThe original denominator disappears
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The visual connects exact notation to nested sets, counts, or a probability tree.
Why does the sample space shrink?
Conditioning on A1 replaces all 380 outcomes with the 190 outcomes in which the first ball is red.
- Learning goalExplain restricted denominator using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionCount the remaining important set
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The visual connects exact notation to nested sets, counts, or a probability tree.
How many favorable outcomes survive?
Inside A1, 10 choices for the first red ball and 9 remaining red choices give 90 outcomes in A1 intersection A2.
- Learning goalExplain restricted numerator using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionNow derive nine nineteenths
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The visual connects exact notation to nested sets, counts, or a probability tree.
How does counting become conditional probability?
P(A2|A1)=90/190=9/19 because the restricted numerator and denominator share a factor of 10.
P(A2|A1)=90/190=9/19 because the restricted numerator and denominator share a factor of 10.
- Learning goalExplain worked result using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionMarginal and conditional probabilities answer different questions
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The visual connects exact notation to nested sets, counts, or a probability tree.
Why is P(A2) not P(A2 | A1)?
Before information P(A2)=10/20=1/2; after a red first draw P(A2|A1)=9/19.
Before information P(A2)=10/20=1/2; after a red first draw P(A2|A1)=9/19.
- Learning goalExplain comparison using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionThe set formula matches the physical story
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The visual connects exact notation to nested sets, counts, or a probability tree.
Where does the formula come from?
P(B|A)=P(A intersection B)/P(A) is the ratio of the overlap to the conditioning set, provided P(A)>0.
P(B|A)=P(A intersection B)/P(A) is the ratio of the overlap to the conditioning set, provided P(A)>0.
- Learning goalExplain formula derivation using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionMultiplication rebuilds the joint event
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The visual connects exact notation to nested sets, counts, or a probability tree.
How do conditional pieces reconnect?
Rearranging gives P(A1 intersection A2)=P(A1)P(A2|A1)=10/20 times 9/19.
Rearranging gives P(A1 intersection A2)=P(A1)P(A2|A1)=10/20 times 9/19.
- Learning goalExplain multiplication rule using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionEvery new condition creates a smaller world
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The visual connects exact notation to nested sets, counts, or a probability tree.
How does a probability chain grow?
For sequential events, each factor is measured inside all information that came before it.
- Learning goalExplain chain rule using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionIndependence is the special case
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The visual connects exact notation to nested sets, counts, or a probability tree.
When does conditioning change nothing?
If A and B are independent, P(B|A)=P(B), so the multiplication rule becomes P(A intersection B)=P(A)P(B).
If A and B are independent, P(B|A)=P(B), so the multiplication rule becomes P(A intersection B)=P(A)P(B).
- Learning goalExplain independence using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionThe multiplication is a sequence of restrictions
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The visual connects exact notation to nested sets, counts, or a probability tree.
Why is multiplying probabilities not blind arithmetic?
Each factor represents entering a smaller valid set; multiplication combines the successive fractions of the original world.
- Learning goalExplain multiplication intuition using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionBreak complex notation into small worlds
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The visual connects exact notation to nested sets, counts, or a probability tree.
How should we solve a long conditional problem?
Name events, state the current world, count or measure the overlap, then multiply only when the chain rule justifies it.
- Learning goalExplain workflow using the restricted-sample-space model.
- Helpful foundationssample space · events · intersection · fractions
- Common trapThe conditioning bar does not mean intersection; it changes the denominator to the known event, while the numerator is the intersection.
- Next connectionApply restricted-world reasoning to Bayes rules and probability trees
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The visual connects exact notation to nested sets, counts, or a probability tree.