The Central Idea
Let
\[ D=\{\text{condition present}\} \]and
\[ D^c=\{\text{condition absent}\}. \]Let
\[ +=\{\text{positive test}\} \]and
\[ -=\{\text{negative test}\}. \]A diagnostic-testing problem compares the hidden state \(D\) or \(D^c\) with the observed result \(+\) or \(-\).
The Four Possible Outcomes
Every tested case falls into one of four categories:
- true positive: \(D\cap+\);
- false negative: \(D\cap-\);
- false positive: \(D^c\cap+\);
- true negative: \(D^c\cap-\).
These four outcomes form a complete classification of test results.
A Diagnostic Table
| Positive Test | Negative Test | |
|---|---|---|
| Condition Present | True Positive | False Negative |
| Condition Absent | False Positive | True Negative |
Many diagnostic quantities are conditional probabilities formed from rows or columns of this table.
Prevalence
The prevalence of the condition is
\[ \boxed{P(D)}. \]It is the proportion of the relevant population that has the condition before the test result is considered.
In Bayesian language, prevalence often serves as the prior probability of the condition.
Sensitivity
Sensitivity is the probability that the test is positive when the condition is present:
\[ \boxed{ \text{Sensitivity}=P(+\mid D) }. \]Sensitivity measures how effectively the test detects positive cases among those who truly have the condition.
True-Positive Rate
Sensitivity is also called the true-positive rate.
Thus,
\[ \boxed{ \text{TPR}=P(+\mid D) }. \]The denominator is the group with the condition.
False-Negative Rate
Among cases with the condition, the test either returns positive or negative.
Therefore,
\[ P(+\mid D)+P(-\mid D)=1. \]The false-negative rate is
\[ \boxed{ \text{FNR}=P(-\mid D)=1-\text{Sensitivity} }. \]Specificity
Specificity is the probability that the test is negative when the condition is absent:
\[ \boxed{ \text{Specificity}=P(-\mid D^c) }. \]Specificity measures how effectively the test correctly rejects negative cases among those who do not have the condition.
True-Negative Rate
Specificity is also called the true-negative rate:
\[ \boxed{ \text{TNR}=P(-\mid D^c) }. \]False-Positive Rate
Among cases without the condition,
\[ P(+\mid D^c)+P(-\mid D^c)=1. \]Therefore, the false-positive rate is
\[ \boxed{ \text{FPR}=P(+\mid D^c)=1-\text{Specificity} }. \]Sensitivity and Specificity Condition on the True State
Sensitivity conditions on
\[ D, \]while specificity conditions on
\[ D^c. \]Both describe test behavior within known truth categories.
They do not directly answer how likely the condition is after a test result is observed.
Positive Predictive Value
The positive predictive value is the probability that the condition is present given a positive test:
\[ \boxed{ \text{PPV}=P(D\mid +) }. \]This reverses the direction of sensitivity.
Negative Predictive Value
The negative predictive value is the probability that the condition is absent given a negative test:
\[ \boxed{ \text{NPV}=P(D^c\mid -) }. \]Predictive Values Condition on the Test Result
PPV conditions on a positive result:
\[ P(D\mid +). \]NPV conditions on a negative result:
\[ P(D^c\mid -). \]These quantities depend on both test characteristics and the prevalence of the condition in the tested population.
Do Not Confuse Sensitivity with Positive Predictive Value
Sensitivity is
\[ P(+\mid D). \]Positive predictive value is
\[ P(D\mid +). \]These probabilities reverse the direction of conditioning and can be very different.
Do Not Confuse Specificity with Negative Predictive Value
Specificity is
\[ P(-\mid D^c). \]Negative predictive value is
\[ P(D^c\mid -). \]Again, the denominators are different.
A Complete Set of Conditional Quantities
Condition on the true state:
\[ P(+\mid D)=\text{sensitivity}, \] \[ P(-\mid D)=\text{false-negative rate}, \] \[ P(-\mid D^c)=\text{specificity}, \] \[ P(+\mid D^c)=\text{false-positive rate}. \]Condition on the observed result:
\[ P(D\mid +)=\text{PPV}, \] \[ P(D^c\mid -)=\text{NPV}. \]A First Diagnostic Example
Suppose a condition has prevalence
\[ P(D)=0.02. \]A test has sensitivity
\[ P(+\mid D)=0.95 \]and specificity
\[ P(-\mid D^c)=0.96. \]Find the Complementary Error Rates
The false-negative rate is
\[ P(-\mid D)=1-0.95=\boxed{0.05}. \]The false-positive rate is
\[ P(+\mid D^c)=1-0.96=\boxed{0.04}. \]Find the Overall Positive-Test Probability
Since
\[ P(D^c)=0.98, \]total probability gives
\[ P(+) = P(D)P(+\mid D) + P(D^c)P(+\mid D^c). \]Therefore,
\[ P(+) = 0.02(0.95)+0.98(0.04). \]Thus,
\[ \boxed{ P(+)=0.0582 }. \]Find the Positive Predictive Value
Bayes' theorem gives
\[ P(D\mid +) = \frac{P(D)P(+\mid D)}{P(+)}. \]Therefore,
\[ P(D\mid +) = \frac{0.02(0.95)}{0.0582} \approx \boxed{0.3265}. \]The positive predictive value is about \(32.65\%\).
Why High Sensitivity Does Not Guarantee High PPV
The test detects \(95\%\) of true cases, but the condition occurs in only \(2\%\) of the population.
Because the group without the condition is much larger, even a \(4\%\) false-positive rate can produce many false positives.
Predictive value therefore depends strongly on prevalence.
Natural Frequencies
The same example can be made concrete by imagining
\[ 10{,}000 \]tested cases.
At \(2\%\) prevalence,
\[ 200 \]have the condition and
\[ 9{,}800 \]do not.
Count the True Positives
Sensitivity is \(95\%\), so among \(200\) cases with the condition,
\[ 0.95(200)=190 \]test positive.
These are the true positives.
Count the False Negatives
The remaining
\[ 200-190=10 \]cases with the condition test negative.
These are the false negatives.
Count the False Positives
The false-positive rate is \(4\%\).
Among \(9{,}800\) cases without the condition,
\[ 0.04(9{,}800)=392 \]test positive.
These are the false positives.
Count the True Negatives
Specificity is \(96\%\), so
\[ 0.96(9{,}800)=9{,}408 \]cases without the condition test negative.
These are the true negatives.
Build the Natural-Frequency Table
| Positive | Negative | Total | |
|---|---|---|---|
| Condition Present | 190 | 10 | 200 |
| Condition Absent | 392 | 9408 | 9800 |
| Total | 582 | 9418 | 10000 |
Read PPV from the Table
Among all positive tests,
\[ 582 \]in total, exactly
\[ 190 \]are true positives.
Therefore,
\[ P(D\mid +) = \frac{190}{582} \approx \boxed{0.3265}. \]Read NPV from the Table
Among all negative tests,
\[ 9418 \]in total,
\[ 9408 \]are true negatives.
Therefore,
\[ P(D^c\mid -) = \frac{9408}{9418} \approx \boxed{0.9989}. \]The negative predictive value is about \(99.89\%\).
The Same Test Can Have Very Different PPV and NPV
In the example,
\[ \text{PPV}\approx32.65\% \]while
\[ \text{NPV}\approx99.89\%. \]The difference arises partly because the condition is rare in the tested population.
The Role of Prevalence
For fixed sensitivity and specificity, changing
\[ P(D) \]changes the predictive values.
In general, increasing prevalence tends to increase positive predictive value and decrease negative predictive value.
Decreasing prevalence tends to have the opposite effect.
Positive Predictive Value Formula
Using sensitivity, specificity, and prevalence,
\[ \boxed{ P(D\mid +) = \frac{P(D)P(+\mid D)} {P(D)P(+\mid D)+P(D^c)P(+\mid D^c)} }. \]Since
\[ P(+\mid D^c)=1-\text{specificity}, \]the formula can be written directly from prevalence, sensitivity, and specificity.
Negative Predictive Value Formula
Similarly,
\[ \boxed{ P(D^c\mid -) = \frac{P(D^c)P(-\mid D^c)} {P(D)P(-\mid D)+P(D^c)P(-\mid D^c)} }. \]Here
\[ P(-\mid D)=1-\text{sensitivity}. \]Example: Higher Prevalence
Keep sensitivity at
\[ 0.95 \]and specificity at
\[ 0.96, \]but suppose prevalence is now
\[ P(D)=0.20. \]Find the New Positive Predictive Value
Since
\[ P(D^c)=0.80, \]the positive probability is
\[ P(+) = 0.20(0.95)+0.80(0.04). \]Thus,
\[ P(+)=0.190+0.032=0.222. \]Therefore,
\[ P(D\mid +) = \frac{0.190}{0.222} \approx \boxed{0.8559}. \]The PPV rises dramatically because the condition is much more common.
Same Test, Different Population
The sensitivity and specificity did not change.
Only the prevalence changed.
Yet the PPV increased from about
\[ 32.65\% \]to about
\[ 85.59\%. \]Predictive values are therefore population-dependent.
Example: Quality-Control Screening
The same mathematics applies outside medicine.
Suppose \(5\%\) of manufactured units are defective:
\[ P(D)=0.05. \]An inspection system flags \(90\%\) of defective units:
\[ P(+\mid D)=0.90. \]It correctly clears \(95\%\) of nondefective units:
\[ P(-\mid D^c)=0.95. \]Find the Flagged-Unit Probability
The false-positive rate is
\[ 1-0.95=0.05. \]Therefore,
\[ P(+) = 0.05(0.90)+0.95(0.05). \]Thus,
\[ P(+)=0.045+0.0475=\boxed{0.0925}. \]Given a Flag, Find the Probability of a Defect
Apply Bayes' theorem:
\[ P(D\mid +) = \frac{0.05(0.90)}{0.0925}. \]Therefore,
\[ \boxed{ P(D\mid +)\approx0.4865 }. \]Under this model, about \(48.65\%\) of flagged units are actually defective.
Accuracy
One overall measure of correct classification is
\[ \boxed{ P(\text{correct}) = P(D\cap+)+P(D^c\cap-) }. \]Using conditional probabilities,
\[ P(\text{correct}) = P(D)P(+\mid D) + P(D^c)P(-\mid D^c). \]Example: Overall Accuracy
Return to the example with
\[ P(D)=0.02, \] \[ P(+\mid D)=0.95, \]and
\[ P(-\mid D^c)=0.96. \]Then
\[ P(\text{correct}) = 0.02(0.95)+0.98(0.96). \]Therefore,
\[ P(\text{correct}) = 0.019+0.9408 = \boxed{0.9598}. \]The overall accuracy is \(95.98\%\).
High Accuracy Can Be Misleading in Imbalanced Populations
If the condition is very rare, a system that almost always predicts “negative” may achieve high overall accuracy while performing poorly at identifying the positive cases.
Therefore, accuracy should not automatically replace sensitivity, specificity, or predictive values.
A Simple Accuracy Warning
Suppose only \(1\%\) of cases are positive.
A classifier that always predicts negative would be correct
\[ 99\% \]of the time.
Yet its sensitivity would be
\[ \boxed{0}. \]A single summary measure can hide important behavior.
False Discovery Rate
The probability that a positive result is false is
\[ \boxed{ P(D^c\mid +) }. \]This is the complement of positive predictive value:
\[ P(D^c\mid +)=1-P(D\mid +). \]It is sometimes called the false discovery rate in classification contexts.
False Omission Probability
The probability that a negative result hides a positive case is
\[ \boxed{ P(D\mid -) }. \]This is the complement of negative predictive value:
\[ P(D\mid -)=1-P(D^c\mid -). \]Positive Likelihood Ratio
A positive result can also be summarized by its likelihood ratio:
\[ \boxed{ LR_+ = \frac{P(+\mid D)} {P(+\mid D^c)} }. \]In terms of sensitivity and specificity,
\[ \boxed{ LR_+ = \frac{\text{sensitivity}} {1-\text{specificity}} }. \]Negative Likelihood Ratio
A negative result has likelihood ratio
\[ \boxed{ LR_- = \frac{P(-\mid D)} {P(-\mid D^c)} }. \]Therefore,
\[ \boxed{ LR_- = \frac{1-\text{sensitivity}} {\text{specificity}} }. \]Interpret the Positive Likelihood Ratio
If
\[ LR_+=10, \]then a positive result is ten times as likely when the condition is present as when it is absent.
This quantity multiplies prior odds into posterior odds after a positive result.
Interpret the Negative Likelihood Ratio
If
\[ LR_-=0.10, \]then a negative result is only one-tenth as likely when the condition is present as when it is absent.
Such evidence reduces the odds in favor of the condition.
Example: Likelihood Ratios from Sensitivity and Specificity
Suppose
\[ \text{sensitivity}=0.90 \]and
\[ \text{specificity}=0.95. \]Then
\[ LR_+ = \frac{0.90}{0.05} = \boxed{18}. \]Also,
\[ LR_- = \frac{0.10}{0.95} \approx \boxed{0.1053}. \]Testing as an Odds Update
If the prior odds of the condition are
\[ O_{\text{prior}}, \]then after a positive test,
\[ \boxed{ O_{\text{post},+} = O_{\text{prior}}LR_+ }. \]After a negative test,
\[ \boxed{ O_{\text{post},-} = O_{\text{prior}}LR_- }. \]Example: Positive Result in Odds Form
Suppose prevalence is
\[ P(D)=0.10. \]Prior odds are
\[ \frac{0.10}{0.90} = \frac{1}{9}. \]Suppose
\[ LR_+=9. \]Then posterior odds are
\[ \frac{1}{9}\cdot9=1. \]Therefore,
\[ \boxed{P(D\mid +)=0.50}. \]Example: Negative Result in Odds Form
Keep prior odds
\[ 1:9 \]and suppose
\[ LR_-=0.10. \]Posterior odds are
\[ \frac{1}{9}(0.10) = \frac{1}{90}. \]Therefore,
\[ P(D\mid -) = \frac{1}{1+90} = \boxed{\frac{1}{91}} \approx \boxed{0.0110}. \]Thresholds Create Tradeoffs
Many diagnostic systems classify a continuous score using a threshold.
Moving the threshold can change:
- sensitivity;
- specificity;
- false-positive rate;
- false-negative rate.
Increasing one performance measure may reduce another.
A Lower Positive Threshold
If a test labels more borderline cases as positive, sensitivity may increase because fewer positive cases are missed.
However, specificity may decrease because more negative cases are also labeled positive.
A Higher Positive Threshold
If a stricter threshold is required for a positive result, specificity may increase because fewer negative cases are falsely labeled positive.
However, sensitivity may decrease because more positive cases are missed.
There Is No Universal Best Threshold
The preferred threshold depends on the consequences of the two error types.
In one application, false negatives may be especially costly.
In another, false positives may create substantial follow-up costs.
Probability quantifies the tradeoff; choosing among tradeoffs requires the surrounding decision context.
Screening and Confirmation
Some systems use a first test designed to identify a broad set of possible positive cases and then apply a more selective second-stage procedure.
Mathematically, this creates a sequential conditional-probability problem.
Each additional result can update the probability again.
Repeated Tests Are Not Automatically Independent
If the same test is repeated, the results may share sources of error.
Therefore, one should not automatically multiply likelihood ratios as though the test results were conditionally independent.
The dependence structure of repeated evidence matters.
Example: A Confusion Matrix
Suppose a classifier is evaluated on \(1{,}000\) cases:
| Predicted Positive | Predicted Negative | Total | |
|---|---|---|---|
| Actually Positive | 180 | 20 | 200 |
| Actually Negative | 80 | 720 | 800 |
| Total | 260 | 740 | 1000 |
Calculate Sensitivity from the Confusion Matrix
Among the \(200\) actually positive cases,
\[ 180 \]are predicted positive.
Therefore,
\[ \boxed{ \text{Sensitivity} = \frac{180}{200} = 0.90 }. \]Calculate Specificity from the Confusion Matrix
Among the \(800\) actually negative cases,
\[ 720 \]are predicted negative.
Therefore,
\[ \boxed{ \text{Specificity} = \frac{720}{800} = 0.90 }. \]Calculate Positive Predictive Value
Among the
\[ 260 \]predicted-positive cases,
\[ 180 \]are actually positive.
Therefore,
\[ \boxed{ \text{PPV} = \frac{180}{260} \approx 0.6923 }. \]Calculate Negative Predictive Value
Among the
\[ 740 \]predicted-negative cases,
\[ 720 \]are actually negative.
Therefore,
\[ \boxed{ \text{NPV} = \frac{720}{740} \approx 0.9730 }. \]Calculate Accuracy from the Confusion Matrix
Correct classifications are
\[ 180+720=900. \]Therefore,
\[ \boxed{ \text{Accuracy} = \frac{900}{1000} = 0.90 }. \]Different Metrics Use Different Denominators
Sensitivity uses the actually positive group.
Specificity uses the actually negative group.
PPV uses the predicted-positive group.
NPV uses the predicted-negative group.
Accuracy uses the entire evaluated population.
Common Error: Assuming a Positive Test Means the Condition Is Present
A positive test does not automatically imply
\[ P(D\mid +)=1. \]False positives may occur.
The correct posterior probability depends on prevalence and test behavior.
Common Error: Equating Sensitivity with PPV
Sensitivity is
\[ P(+\mid D). \]PPV is
\[ P(D\mid +). \]Reversing the condition changes the denominator and usually changes the probability.
Common Error: Equating Specificity with NPV
Specificity is
\[ P(-\mid D^c). \]NPV is
\[ P(D^c\mid -). \]These are not interchangeable.
Common Error: Using Specificity as the False-Positive Rate
Specificity is
\[ P(-\mid D^c). \]The false-positive rate is its complement:
\[ \boxed{ P(+\mid D^c)=1-\text{specificity} }. \]Common Error: Ignoring Prevalence
Sensitivity and specificity alone do not determine PPV or NPV.
Predictive values also depend on
\[ P(D). \]The same test can have different predictive values in populations with different prevalence.
Common Error: Treating Accuracy as the Only Performance Measure
Accuracy can conceal important asymmetries between false positives and false negatives.
This is especially important when one class is rare.
Common Error: Multiplying Repeated-Test Evidence Without a Dependence Model
Repeated test outcomes may not be conditionally independent.
If their errors are correlated, multiplying separate likelihood ratios can overstate the amount of new information.
Common Error: Confusing Mathematical Probability with a Decision
A posterior probability quantifies uncertainty under a model.
It does not by itself determine what action should be taken.
Decisions may also depend on costs, benefits, risks, policies, thresholds, and domain-specific judgment.
A Reliable Procedure from Sensitivity and Specificity
- Define \(D\), \(D^c\), \(+\), and \(-\).
- Write the prevalence \(P(D)\).
- Compute \(P(D^c)=1-P(D)\).
- Write sensitivity \(P(+\mid D)\).
- Write specificity \(P(-\mid D^c)\).
- Compute the false-positive and false-negative rates by complements.
- Use total probability to find \(P(+)\) or \(P(-)\).
- Use Bayes' theorem to calculate the desired predictive value.
- Check the result against a natural-frequency table when useful.
A Reliable Procedure from Counts
- Build a \(2\times2\) table of actual state versus test result.
- Check that row and column totals agree with the population total.
- Use row totals for sensitivity and specificity.
- Use column totals for predictive values.
- Use all cases for overall accuracy.
- Translate every ratio back into a conditional probability statement.
Example: Full Workflow
Suppose
\[ P(D)=0.10, \]sensitivity is
\[ 0.85, \]and specificity is
\[ 0.90. \]Then
\[ P(D^c)=0.90, \] \[ P(+\mid D^c)=0.10. \]The overall positive probability is
\[ P(+) = 0.10(0.85)+0.90(0.10). \]Therefore,
\[ P(+)=0.085+0.090=0.175. \]The positive predictive value is
\[ P(D\mid +) = \frac{0.085}{0.175} = \boxed{\frac{17}{35}} \approx \boxed{0.4857}. \]Practice 1: Complementary Error Rates
A test has sensitivity \(0.92\) and specificity \(0.95\).
Find the false-negative rate and false-positive rate.
Practice 2: Positive Predictive Value
Suppose
\[ P(D)=0.10, \] \[ P(+\mid D)=0.90, \]and specificity is
\[ 0.95. \]Find \(P(D\mid +)\).
Practice 3: Negative Predictive Value
Using the same information as Practice 2, find
\[ P(D^c\mid -). \]Practice 4: Natural Frequencies
A condition affects \(5\%\) of a population. A test has sensitivity \(80\%\) and specificity \(90\%\).
In a hypothetical population of \(1{,}000\), find the expected numbers of:
- true positives;
- false negatives;
- false positives;
- true negatives.
Practice 5: Confusion Matrix
A classifier produces:
- \(90\) true positives;
- \(10\) false negatives;
- \(40\) false positives;
- \(360\) true negatives.
Find sensitivity and specificity.
Practice 6: Predictive Values from Counts
Using the counts from Practice 5, find PPV and NPV.
Practice 7: Accuracy
Using the counts from Practice 5, find the overall accuracy.
Practice 8: Positive Likelihood Ratio
A test has sensitivity
\[ 0.90 \]and specificity
\[ 0.80. \]Find \(LR_+\).
Practice 9: Negative Likelihood Ratio
Using the same test as Practice 8, find \(LR_-\).
Practice 10: Identify the Error
A test has sensitivity \(99\%\). A student concludes that anyone with a positive result has a \(99\%\) probability of having the condition.
Explain the error.
Practice Answers
Practice 1
The false-negative rate is
\[ 1-0.92=\boxed{0.08}. \]The false-positive rate is
\[ 1-0.95=\boxed{0.05}. \]Practice 2
The false-positive rate is
\[ 0.05. \]Therefore,
\[ P(+) = 0.10(0.90)+0.90(0.05) = 0.09+0.045 = 0.135. \]Thus,
\[ P(D\mid +) = \frac{0.09}{0.135} = \boxed{\frac{2}{3}} \approx \boxed{0.6667}. \]Practice 3
The false-negative rate is
\[ 0.10. \]Therefore,
\[ P(-) = 0.10(0.10)+0.90(0.95) = 0.01+0.855 = 0.865. \]Thus,
\[ P(D^c\mid -) = \frac{0.90(0.95)}{0.865} = \frac{0.855}{0.865} \approx \boxed{0.9884}. \]Practice 4
Out of \(1{,}000\), the expected number with the condition is
\[ 50, \]and without it is
\[ 950. \]True positives:
\[ 0.80(50)=\boxed{40}. \]False negatives:
\[ 50-40=\boxed{10}. \]True negatives:
\[ 0.90(950)=\boxed{855}. \]False positives:
\[ 950-855=\boxed{95}. \]Practice 5
Sensitivity:
\[ \frac{90}{90+10} = \boxed{0.90}. \]Specificity:
\[ \frac{360}{360+40} = \boxed{0.90}. \]Practice 6
PPV:
\[ \frac{90}{90+40} = \frac{90}{130} \approx \boxed{0.6923}. \]NPV:
\[ \frac{360}{360+10} = \frac{360}{370} \approx \boxed{0.9730}. \]Practice 7
Correct classifications:
\[ 90+360=450. \]Total classifications:
\[ 90+10+40+360=500. \]Therefore,
\[ \boxed{ \text{Accuracy}=\frac{450}{500}=0.90 }. \]Practice 8
\[ LR_+ = \frac{0.90}{1-0.80} = \frac{0.90}{0.20} = \boxed{4.5}. \]Practice 9
\[ LR_- = \frac{1-0.90}{0.80} = \frac{0.10}{0.80} = \boxed{0.125}. \]Practice 10
Sensitivity is
\[ P(+\mid D), \]while the probability that the condition is present after a positive result is
\[ P(D\mid +). \]These are different conditional probabilities. To find the second, we also need the prevalence and the false-positive behavior of the test.
What to Remember
Diagnostic testing uses four basic outcomes:
- true positive;
- false negative;
- false positive;
- true negative.
The central conditional probabilities are
\[ \boxed{ \text{Sensitivity}=P(+\mid D) } \] \[ \boxed{ \text{Specificity}=P(-\mid D^c) } \] \[ \boxed{ \text{PPV}=P(D\mid +) } \] \[ \boxed{ \text{NPV}=P(D^c\mid -) }. \]- False-negative rate is \(1-\) sensitivity.
- False-positive rate is \(1-\) specificity.
- Predictive values depend on prevalence.
- Bayes' theorem reverses test-performance probabilities into post-test probabilities.
- Natural frequencies often make the calculation easier to interpret.
- Accuracy alone can hide poor performance on a rare class.
- Likelihood ratios summarize how strongly a test result shifts prior odds.
- Repeated tests require attention to dependence between results.
The central habit is: identify the denominator first—true state for sensitivity and specificity, observed result for predictive values.
Where We Go Next
Diagnostic testing applies Bayesian reasoning to one of its most familiar two-state settings. The same ideas extend beyond tests to expectations of numerical outcomes when information is known.
The next lesson, Conditional Expectation, develops the expected value of a random variable within a conditioned event or information set and connects probability updating with averages under restricted information.