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{{T}}
[[Category:English]]
[[Category:Conditional Probability]]

{{BR}}
= Introduction =

'''Conditional probability''' describes how the probability of an event changes when you know that another event has occurred. It is one of the central ideas of [[English:Probability|probability theory]] because information changes the set of outcomes that remain possible. In Grades 11–13, conditional probability connects counting, fractions, [[English:Venn diagram|Venn diagrams]], [[English:Tree diagram|probability trees]], [[English:Contingency table|two-way tables]], [[English:Independence (probability theory)|independence]], and [[English:Bayes' theorem|Bayes' theorem]].

By the end of this aiMOOC, you should be able to interpret conditional-probability notation, calculate conditional probabilities from data and models, construct and read tree diagrams, test whether events are independent, reverse a condition with Bayes' theorem, and explain how base rates affect real-world decisions.

[[File:Conditional probability.svg|500px|frameless|center]]

The diagram above shows a sample space with overlapping events. Conditioning means restricting your attention to the event that is known to have occurred and then asking what fraction of that restricted probability belongs to the event of interest.

{{#ev:youtube|https://www.youtube.com/watch?v=_IgyaD7vOOA|500|center}}

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= Core Ideas =

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== Events, Intersections, and the Reduced Sample Space ==

An '''event''' is a set of outcomes. If event A means “the selected student studies mathematics” and event B means “the selected student is in Grade 12,” then <math>A\cap B</math> means that both statements are true. When you are told that B has happened, outcomes outside B are no longer relevant to the conditional question.

This is why the word '''given''' matters. The notation <math>P(A\mid B)</math> is read as “the probability of A given B.” The vertical bar does not mean division. It announces the condition that defines the reduced sample space.

For equally likely outcomes, you can think of conditional probability as a ratio of counts:

<math>P(A\mid B)=\frac{\text{number of outcomes in }A\cap B}{\text{number of outcomes in }B}</math>.

This count interpretation is especially useful when working with cards, dice, survey data, or classroom datasets.

[[File:Conditional probability venn 1-10.svg|500px|frameless|center]]

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== The Definition of Conditional Probability ==

For events A and B with <math>P(B)>0</math>,

<math>P(A\mid B)=\frac{P(A\cap B)}{P(B)}</math>.

The denominator is the probability of the event you are '''given'''. The numerator is the probability that both the event of interest and the given event occur.

'''Example.''' Suppose <math>P(A\cap B)=0.18</math> and <math>P(B)=0.30</math>. Then

<math>P(A\mid B)=\frac{0.18}{0.30}=0.60</math>.

The condition has changed the reference set: instead of comparing the intersection with the whole sample space, you compare it with B.

A conditional probability is only defined by this formula when the conditioning event has positive probability. At this level, problems are normally designed so that <math>P(B)>0</math>.

{{BR}}
== Reading Two-Way Tables ==

A [[English:Contingency table|two-way table]] organizes counts for two categorical variables. It is one of the clearest ways to distinguish joint, marginal, and conditional information.

Suppose 200 students are classified by whether they take an advanced mathematics course and whether they participate in a science club:

{| class="wikitable" style="margin:auto"
!
! Science club
! Not in science club
! Total
|-
! Advanced mathematics
| 54
| 36
| 90
|-
! Not advanced mathematics
| 26
| 84
| 110
|-
! Total
| 80
| 120
| 200
|}

If A is “advanced mathematics” and S is “science club,” then

<math>P(A\mid S)=\frac{54}{80}=0.675</math>.

The denominator is 80 because the condition S restricts attention to science-club members. By contrast,

<math>P(S\mid A)=\frac{54}{90}=0.60</math>.

These two conditional probabilities use the same intersection count but different denominators. In general, <math>P(A\mid B)</math> and <math>P(B\mid A)</math> are not equal.

[[File:Contingency Table.png|500px|frameless|center]]

{{BR}}
== The Multiplication Rule ==

Rearranging the definition gives the '''multiplication rule''':

<math>P(A\cap B)=P(B)P(A\mid B)</math>.

Equivalently,

<math>P(A\cap B)=P(A)P(B\mid A)</math>.

This rule is useful when a process unfolds in stages. You multiply along a path because reaching the end of that path requires each stage on the path to occur.

'''Example.''' A school randomly selects one student. The probability that the student is in Grade 12 is 0.40. Among Grade 12 students, the probability of studying physics is 0.35. Therefore,

<math>P(\text{Grade 12 and physics})=0.40\cdot0.35=0.14</math>.

{{BR}}
== Probability Trees ==

A probability tree represents sequential or conditional structure. The probabilities on branches leaving the same node must add to 1. A branch probability after the first stage is conditional on the event represented by the path that led to that node.

[[File:Probability tree diagram.svg|500px|frameless|center]]

To find the probability of a complete path, multiply the branch probabilities along that path. To find the probability of an event that can happen through several mutually exclusive paths, add the probabilities of those paths.

{{#ev:youtube|https://www.youtube.com/watch?v=hxEdXUB_IdQ|500|center}}

'''Tree example.''' A digital filter flags 8% of incoming messages. Among flagged messages, 70% actually contain the target feature. The probability that a randomly chosen message is both flagged and contains the target feature is

<math>0.08\cdot0.70=0.056</math>.

A tree diagram helps you keep the direction of each conditional probability visible.

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= Independence and Dependence =

{{BR}}
== Testing Independence ==

Events A and B are '''independent''' when knowing that one occurred does not change the probability of the other. If the relevant probabilities are positive, any of the following equivalent conditions can be used:

<math>P(A\mid B)=P(A)</math>

<math>P(B\mid A)=P(B)</math>

<math>P(A\cap B)=P(A)P(B)</math>.

If these equalities do not hold, the events are dependent.

'''Example.''' If <math>P(A)=0.50</math>, <math>P(B)=0.40</math>, and <math>P(A\cap B)=0.20</math>, then <math>P(A)P(B)=0.20</math>. The events are independent.

Do not confuse '''independent''' with '''mutually exclusive'''. If two nonzero-probability events are mutually exclusive, they cannot be independent because the occurrence of one makes the other impossible.

{{#ev:youtube|https://www.youtube.com/watch?v=pIfpHdGVwLU|500|center}}

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== Independence in Repeated Experiments ==

Repeated trials are often modeled as independent, but independence is an assumption that must be justified by the situation. For example, repeated tosses of an ideal coin are modeled as independent because the result of one toss does not alter the probability distribution of the next toss.

Sampling without replacement is usually dependent because the composition of the remaining population changes after each selection. Sampling with replacement can restore the same probability distribution from draw to draw.

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= Bayes' Theorem and Reversing Conditions =

{{BR}}
== Why Reversing a Condition Is Difficult ==

A common error is to treat <math>P(A\mid B)</math> as if it were the same as <math>P(B\mid A)</math>. The first asks about A inside the restricted world where B is known. The second asks about B inside a different restricted world where A is known.

The two probabilities are connected, but they are not generally equal. Bayes' theorem provides the correct relationship.

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== Bayes' Theorem ==

For <math>P(B)>0</math>,

<math>P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}</math>.

The theorem follows because the joint probability can be written in two ways:

<math>P(A\cap B)=P(A\mid B)P(B)=P(B\mid A)P(A)</math>.

Solving this equality for <math>P(A\mid B)</math> gives Bayes' theorem.

[[File:Bayes theorem visualisation.svg|500px|frameless|center]]

The language of Bayesian reasoning often distinguishes a '''prior probability''' before new evidence, a '''likelihood''' describing how compatible the evidence is with a hypothesis, and a '''posterior probability''' after the evidence has been incorporated.

{{#ev:youtube|https://www.youtube.com/watch?v=HZGCoVF3YvM|500|center}}

{{BR}}
== Law of Total Probability ==

Bayes' theorem often requires the denominator <math>P(B)</math>. When A and its complement <math>A^c</math> split the sample space into two non-overlapping cases,

<math>P(B)=P(B\mid A)P(A)+P(B\mid A^c)P(A^c)</math>.

More generally, if mutually exclusive events <math>A_1,\ldots,A_n</math> partition the sample space, then

<math>P(B)=\sum_i P(B\mid A_i)P(A_i)</math>.

This is the '''law of total probability'''. It combines all distinct ways in which B can occur.

[[File:Bayes theorem tree diagrams.svg|500px|frameless|center]]

{{BR}}
== A Base-Rate Example ==

Imagine a quality-control system in which 2% of products are defective. A scanner flags 90% of defective products and incorrectly flags 5% of non-defective products.

Let D mean “defective” and F mean “flagged.” Then

<math>P(D)=0.02</math>, <math>P(F\mid D)=0.90</math>, and <math>P(F\mid D^c)=0.05</math>.

First compute the total probability of being flagged:

<math>P(F)=0.90\cdot0.02+0.05\cdot0.98=0.067</math>.

Then use Bayes' theorem:

<math>P(D\mid F)=\frac{0.90\cdot0.02}{0.067}\approx0.269</math>.

So a flagged product has about a 26.9% probability of actually being defective under this model. The result may feel surprising because the defect rate is low. This illustrates why the '''base rate''' matters when evidence is interpreted.

A natural-frequency version can make the same idea intuitive. Out of 10,000 products, about 200 are defective; about 180 of those are flagged. Of the 9,800 non-defective products, about 490 are falsely flagged. Among the 670 flagged products, 180 are defective, giving <math>180/670\approx0.269</math>.

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= Representations and Problem-Solving Strategies =

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== Choosing a Representation ==

Different representations highlight different structures.

# [[English:Venn diagram|Venn diagram]]: Useful for visualizing sets, intersections, unions, and a restricted sample space.
# [[English:Contingency table|Contingency table]]: Useful when the problem gives counts or percentages for two categorical variables.
# [[English:Tree diagram|Tree diagram]]: Useful for staged processes, sequential events, and reverse-conditioning problems.
# [[English:Bayes' theorem|Bayes' theorem]]: Useful when you know a probability in one direction and need the reversed conditional probability.

The best representation is the one that makes the given condition and the relevant denominator unmistakable.

{{BR}}
== A Reliable Solution Routine ==

When you solve a conditional-probability problem, first define the events in words. Next identify the event after the word “given”; that event determines the conditioning set. Then choose a table, tree, Venn diagram, or formula. Check whether the denominator is the probability or count of the conditioning event. Finally, interpret the numerical answer in the context of the original question.

A probability must lie between 0 and 1. If your result does not, the setup or arithmetic is wrong. You should also ask whether the answer is plausible: did the new information make the target event more likely, less likely, or leave it unchanged?

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= Common Misconceptions =

'''Mistake 1: Reversing the condition.''' <math>P(A\mid B)</math> is not generally equal to <math>P(B\mid A)</math>. Read the notation in words before calculating.

'''Mistake 2: Using the whole sample space as the denominator.''' Once you condition on B, the denominator must refer to B.

'''Mistake 3: Assuming independence without evidence.''' Independence is a mathematical property or modeling assumption, not a default rule.

'''Mistake 4: Confusing independence with mutual exclusivity.''' Nonzero-probability mutually exclusive events are dependent because knowing one occurred tells you the other did not.

'''Mistake 5: Ignoring base rates.''' A high likelihood such as <math>P(F\mid D)</math> does not by itself determine the reversed probability <math>P(D\mid F)</math>.

'''Mistake 6: Adding branch probabilities along a single tree path.''' Along one path you multiply; across mutually exclusive complete paths you add.

{{BR}}
= Applications =

Conditional probability is used whenever decisions depend on partial information. In [[English:Data science|data science]], it appears in classification, prediction, and probabilistic models. In engineering, it supports reliability analysis and fault detection. In economics and finance, it helps model decisions under uncertainty. In quality control, it connects test outcomes with defect rates. In science, it helps researchers update interpretations when new evidence becomes available.

At school level, the most important goal is not merely to apply a formula. You should be able to explain which event is being conditioned on, why a denominator is appropriate, how a representation encodes the information, and what the answer means in context.

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= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

'''If P(A intersect B) is 0.18 and P(B) is 0.30, what is the value of P(A given B)?'''
(0.60)
(!0.12)
(!0.30)
(!0.48)

{{E}}
<br>

{{MC}}

'''Which event determines the denominator when calculating the probability of A given B?'''
(The given event B)
(!The target event A)
(!The union of A and B)
(!The complement of B)

{{E}}
<br>

{{MC}}

'''What does it mean if the probability of A given B equals the probability of A?'''
(A and B are independent)
(!A and B are mutually exclusive)
(!A is impossible)
(!B is certain)

{{E}}
<br>

{{MC}}

'''If two nonzero-probability events are mutually exclusive, what is true?'''
(They are dependent)
(!They are independent)
(!They are identical)
(!They both have probability one)

{{E}}
<br>

{{MC}}

'''In a probability tree, how do you find the probability of one complete path?'''
(Multiply the branch probabilities)
(!Add the branch probabilities)
(!Subtract the final branch)
(!Divide every branch by two)

{{E}}
<br>

{{MC}}

'''A table contains 80 science-club students, and 54 of them take advanced mathematics. What is the probability of advanced mathematics given science-club membership?'''
(0.675)
(!0.450)
(!0.600)
(!0.800)

{{E}}
<br>

{{MC}}

'''Which theorem is used to reverse the direction of a conditional probability?'''
(Bayes theorem)
(!Pythagorean theorem)
(!Binomial theorem)
(!Fundamental theorem of calculus)

{{E}}
<br>

{{MC}}

'''Which quantity is especially important when interpreting a positive flag for a rare event?'''
(The base rate)
(!The sample label)
(!The drawing scale)
(!The order of the alphabet)

{{E}}
<br>

{{MC}}

'''When sampling without replacement, why are successive draws usually dependent?'''
(The population composition changes)
(!Every outcome becomes equally likely)
(!The sample space becomes infinite)
(!Probabilities stop being numerical)

{{E}}
<br>

{{MC}}

'''What must be true for the elementary formula for the probability of A given B to be used?'''
(The probability of B is positive)
(!The probability of A is zero)
(!A and B are disjoint)
(!A and B are identical)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| Conditional probability || Probability of an event under a stated condition
|-
| Intersection || Event in which two event conditions are both satisfied
|-
| Independence || Property that knowledge of one event does not change the probability of another
|-
| Prior || Probability assigned before the new evidence is incorporated
|-
| Posterior || Probability after the evidence has been incorporated
|-
| Base rate || Background frequency of an event in the relevant population
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! Match the correct terms.
! Topic
|-
| '''Probability inside a restricted condition'''
| Conditional probability
|-
| '''Probability that both events occur'''
| Joint probability
|-
| '''No change in probability after conditioning'''
| Independence
|-
| '''Reverse a conditional direction using prior information'''
| Bayes theorem
|-
| '''Multiply probabilities along successive branches'''
| Tree diagram
|}
{{E}}

<br>
...
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| Conditioning || What process restricts the sample space to a known event?
|-
| Intersection || What one-word term describes the event in which A and B both occur?
|-
| Independence || What property means that knowing one event does not change the probability of another?
|-
| Posterior || What is the updated probability after evidence has been incorporated?
|-
| Likelihood || What term describes how compatible observed evidence is with a hypothesis in Bayesian reasoning?
|-
| Contingency || What word completes the phrase two-way blank table used for categorical counts?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Conditional+Probability </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
Conditional probability asks about an event under a stated { condition }. The event after the word given defines the reduced { sample space }. The numerator in the standard formula is the probability of the { intersection }. In a tree diagram, probabilities along one complete path are { multiplied }. If conditioning does not change the probability of an event, the events are { independent }. Bayes' theorem connects a reversed conditional probability with a prior and a { likelihood }. The background frequency of an event is called its { base rate }. When several mutually exclusive cases can produce the same observation, the law of total probability combines those { cases }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Conditional probability diary|Conditional probability diary]]: Find four everyday statements that use words such as given, if, among, or of those; rewrite each statement using event notation and explain the conditioning event.
# [[English:Two-way table project|Two-way table project]]: Create a small anonymous classroom survey with two non-sensitive categorical variables, organize the results in a two-way table, and calculate two conditional probabilities in opposite directions.
# [[English:Venn diagram explanation|Venn diagram explanation]]: Draw or digitally create a Venn diagram for two overlapping events and record a short explanation of how the picture changes when you condition on one event.
# [[English:Probability video|Probability video]]: Produce a one-minute video that explains why the denominator in a conditional probability comes from the given event.

{{BR}}
=== Standard ===
# [[English:Probability tree experiment|Probability tree experiment]]: Run a two-stage random experiment with coins, cards, or a digital simulator, build a probability tree, compare theoretical and experimental conditional probabilities, and discuss differences.
# [[English:Independence investigation|Independence investigation]]: Design a dataset or simulation for two events, test independence in two different mathematical ways, and explain whether the evidence supports an independence model.
# [[English:Base-rate interview|Base-rate interview]]: Interview at least two people about a fictional screening or quality-control scenario, record their intuitive estimates, then compare those estimates with a correct natural-frequency calculation.
# [[English:Media critique in probability|Media critique in probability]]: Find a news, advertising, or social-media claim that uses a conditional percentage, identify the conditioning group, and rewrite the claim so the direction of the probability is unambiguous.

{{BR}}
=== Advanced ===
# [[English:Bayes theorem model|Bayes theorem model]]: Build a spreadsheet or coded simulation for a rare-event detection problem, vary the prior probability, and graph how the posterior probability changes while test characteristics stay fixed.
# [[English:Conditional probability research poster|Conditional probability research poster]]: Create a research poster connecting conditional probability to one field such as data science, genetics, reliability engineering, economics, or machine learning, with a worked example and sources.
# [[English:Probability learning visit|Probability learning visit]]: Visit a library, university mathematics department, data-science lab, science museum, or suitable virtual institution and document one real application of conditional reasoning in a photo essay or illustrated report.
# [[English:Monte Carlo comparison|Monte Carlo comparison]]: Write or adapt a simulation that estimates a conditional probability, compare the estimate with an exact calculation, investigate convergence as the number of trials increases, and explain the role of random variation.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Model selection assessment|Model selection assessment]]: Given one scenario in table form and one in sequential form, choose suitable representations, justify each choice, and calculate the requested conditional probabilities.
# [[English:Reasoning with independence|Reasoning with independence]]: Analyze a claim that two events are independent, verify or refute it using numerical evidence, and explain why mutual exclusivity would be a different relationship.
# [[English:Bayesian transfer task|Bayesian transfer task]]: Solve a new reverse-conditioning problem, first with a tree or natural frequencies and then with Bayes' theorem, and explain why the two methods agree.
# [[English:Error analysis|Error analysis]]: Inspect a worked solution that uses the wrong denominator or reverses a conditional probability, identify the exact reasoning error, repair the solution, and state a rule that prevents the mistake.
# [[English:Base-rate decision task|Base-rate decision task]]: Compare two detection systems that have different false-positive rates and are used in populations with different base rates; determine which posterior probabilities matter for a stated decision.
# [[English:Communication assessment|Communication assessment]]: Write a short explanation for a non-specialist that distinguishes joint, marginal, and conditional probability using one coherent example without relying only on formulas.

<br>
<br>

{{BR}}
= Evidence of Learning =

Strong evidence of learning includes both correct calculations and clear reasoning.

'''Knowledge:''' You can define conditional probability, joint probability, independence, prior probability, likelihood, posterior probability, and base rate, and you can state the multiplication rule, the law of total probability, and Bayes' theorem.

'''Skills:''' You can translate between verbal descriptions, formulas, two-way tables, Venn diagrams, probability trees, and natural frequencies. You can identify the conditioning event, select a valid denominator, test independence, and check whether an answer is mathematically and contextually plausible.

'''Products:''' Useful evidence may include a solved problem set, annotated probability tree, two-way table, simulation, spreadsheet model, research poster, explanatory video, or written critique of a probability claim.

'''Transfer:''' You can apply conditional reasoning to an unfamiliar situation, explain how base rates affect conclusions, compare alternative probabilistic models, and communicate uncertainty without reversing conditions.

<br>
<br>

{{BR}}
= OERs on the Topic =

<iframe> https://en.m.wikipedia.org/wiki/Conditional_probability </iframe>

For an open-textbook treatment of probability and Bayes' theorem, see [https://openstax.org/books/principles-data-science/pages/3-4-probability-theory OpenStax: Probability Theory].

<br>

{{BR}}
= Linked Learning Areas =

{| align=center
{{:D-Tab}}
'''[[English:Conditional Probability|Conditional Probability]]'''
# [[English:Probability|Probability]]
# [[English:Sample space|Sample space]]
# [[English:Event (probability theory)|Event]]
# [[English:Intersection (set theory)|Intersection]]
# [[English:Contingency table|Contingency table]]
# [[English:Tree diagram|Tree diagram]]
# [[English:Independence (probability theory)|Independence]]
# [[English:Bayes' theorem|Bayes' theorem]]
# [[English:Law of total probability|Law of total probability]]
# [[English:Statistics|Statistics]]
# [[English:Data science|Data science]]
|}

[[Category:English]]
[[Category:Conditional Probability]]
[[Category:Mathematics]]
[[Category:Probability]]
[[Category:Statistics]]
[[Category:Grades 11-13]]
[[Category:Secondary education]]
[[Category:Data science]]

{{BR}}
= aiMOOC Projects =
[[Category:AI_MOOC]] [[Category:GPT aiMOOC]]
{{MT}}

English:Conditional Probability - Versionsgeschichte

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