Person 1: True Positive (Sick + Correct Test)

Conditional Probability

Understanding how probabilities change when new information is available, including the Multiplication Rule, Independence, the Theorem of Total Probability, and Bayes' Theorem.

In engineering, we rarely evaluate risks in isolation. Our assessment of a structure's failure probability changes if we already know that a defect was present during construction. This updating of probabilities based on prior knowledge is called conditional probability.

The Concept of Conditional Probability

Adjusting the probability of an event given that another event has already occurred.

Conditional Probability, $P (B ∣ A)$

The probability that event $B$ occurs, given that event $A$ has already occurred. It effectively reduces the sample space from $S$ to just the outcomes in $A$ .

P(B|A) = \frac{P(A \cap B)}{P(A)} \quad \text{for } P(A) > 0

The Multiplication Rule (General)

Used to find the probability that two events $A$ and $B$ occur simultaneously ( $A \cap B$ ). It is a direct rearrangement of the conditional probability formula.

P(A \cap B) = P(A) \cdot P(B|A)

Or, symmetrically:

P(A \cap B) = P(B) \cdot P(A|B)

Independent Events

When the occurrence of one event has absolutely no effect on the probability of another.

In structural engineering, deciding whether the failure of one column affects the failure probability of an adjacent column is a critical assessment of independence.

Independence

Two events $A$ and $B$ are independent if and only if the knowledge that $A$ occurred does not change the probability of $B$ occurring. Mathematically, this is expressed as:

P(B|A) = P(B) \quad \text{and} \quad P(A|B) = P(A)

Substituting this into the general multiplication rule yields the Special Multiplication Rule for Independent Events:

P(A \cap B) = P(A) \cdot P(B)

Independence of Multiple Events

For three events $A, B, C$ to be mutually independent, they must be pairwise independent ( $P(A \cap B) = P(A)P(B)$ , etc.), AND their joint probability must equal the product of their individual probabilities: $P(A \cap B \cap C) = P(A)P(B)P(C)$ .

The Theorem of Total Probability and Bayes' Theorem

Advanced tools for calculating probabilities when an event can occur through multiple, distinct pathways.

The Theorem of Total Probability

If a sample space $S$ is partitioned into $n$ mutually exclusive and collectively exhaustive events ( $B_1, B_2, \dots, B_n$ ), then the probability of any event $A$ occurring is the sum of the probabilities of $A$ occurring in conjunction with each partition.

P(A) = \sum_{i=1}^{n} P(B_i \cap A) = \sum_{i=1}^{n} P(B_i) \cdot P(A|B_i)

Engineering Application: If a concrete batch $A$ can fail due to poor mixing ( $B_1$ ), improper curing ( $B_2$ ), or bad aggregates ( $B_3$ ), the total probability of failure $P(A)$ is the sum of the probabilities of failure given each specific cause, weighted by how frequently each cause occurs.

Bayes' Theorem

A powerful mathematical formula used to update the probabilities of hypotheses when given new evidence. It essentially reverses the conditional probability. If we know the probability of an effect given a cause ( $P(A|B_i)$ ), Bayes' Theorem lets us find the probability of a specific cause given that the effect occurred ( $P(B_i|A)$ ).

P(B_i|A) = \frac{P(B_i) \cdot P(A|B_i)}{\sum_{j=1}^{n} P(B_j) \cdot P(A|B_j)}

Prior Probability ( $P(B_i)$ ): Our initial estimate of the probability of cause $B_i$ before observing the evidence $A$ .
Likelihood ( $P(A|B_i)$ ): The probability of observing evidence $A$ if cause $B_i$ is true.
Posterior Probability ( $P(B_i|A)$ ): The updated probability of cause $B_i$ after observing the evidence $A$ .

Interact with the simulation below to explore conditional probability and Bayes' Theorem.

Engineering Data Analysis

Bayes' Theorem & Diagnostic Testing Explorer

Prior / Base Rate

P(D)

(Defective)2.0%

The prior probability of a random component being defective.

Sensitivity / True Positive

P(T|D)

95.0%

The probability that the test is positive given that a defect is present.

False Alarm / False Positive

P(T|G)

10.0%

The probability that the test flag is positive when NO defect is present.

Posterior Probability Calculation

16.2%

$P(D|T)$ — Probability that a component is actually defective given a positive test flag.

Total Positives

P(T)

11.7%

True Positives

P(T|D)P(D)

1.90%

False Positives

P(T|G)P(G)

9.80%

Explore the application of Bayes' Theorem to diagnostic testing by adjusting prevalence, sensitivity, and specificity in the simulation below.

Engineering Data Analysis • Topic 4

Bayes' Theorem in Diagnostic Testing

Prevalence

P(D)

10%

Sensitivity

P(+\mid D)

90%

Specificity

P(-\mid D^c)

90%

Population Grid (N = 100)

True Positive (TP: 9)

False Negative (FN: 1)

False Positive (FP: 9)

True Negative (TN: 81)

PPV (Post-test prob of disease | positive test)

50.0%

P(D\mid +) = \frac{P(+\mid D)P(D)}{P(+\mid D)P(D) + P(+\mid D^c)P(D^c)}

NPV (Post-test prob of healthy | negative test)

98.8%

P(D^c\mid -) = \frac{P(-\mid D^c)P(D^c)}{P(-\mid D^c)P(D^c) + P(-\mid D)P(D)}

Key Takeaways

Conditional Probability ( $P(B|A)$ ): The probability of $B$ given that $A$ has occurred; shrinks the sample space to $A$ .
Multiplication Rule: Used for "A AND B" scenarios ( $A \cap B$ ).
Independence: If $A$ and $B$ are independent, $P(A \cap B) = P(A) \cdot P(B)$ .
Theorem of Total Probability: Useful for finding the total probability of an event that can occur across multiple mutually exclusive pathways.
Bayes' Theorem: Allows engineers to "work backward" from an observed effect (e.g., structural failure) to determine the most likely cause, updating prior beliefs with new evidence.

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Conditional Probability

The Concept of Conditional Probability

Conditional Probability, P(B∣A)P(B|A)P(B∣A)