3.1 Conditional probability and Bayes' theorem

Conditional probability helps engineers make informed decisions by calculating the likelihood of events given prior occurrences. It's crucial for analyzing complex systems and updating probabilities based on new information, enabling better risk assessment and decision-making in uncertain situations.

Bayes' theorem takes this a step further, allowing engineers to update probabilities when new evidence emerges. This powerful tool is essential in various fields, from fault diagnosis to quality control, helping refine predictions and improve system reliability.

Conditional Probability

Conditional probability in engineering

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• Probability of an event A occurring given that another event B has already occurred, denoted as $[P(A|B)](https://www.fiveableKeyTerm:p(a|b))$
• Calculated using the formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $P(A \cap B)$ represents the probability of both events A and B occurring simultaneously
• Allows for updating probabilities based on new information or evidence (weather forecasting, medical diagnosis)
• Enables modeling and analyzing complex systems with dependent events (power grid reliability, manufacturing processes)
• Facilitates informed decision-making under uncertainty (project risk assessment, investment strategies)

Application of conditional probability formula

• Identify the events of interest and their dependencies (component failures, quality control inspections)
• Determine the probability of the conditioning event, $P(B)$
• Calculate the probability of both events occurring simultaneously, $P(A \cap B)$
• Apply the conditional probability formula, $P(A|B) = \frac{P(A \cap B)}{P(B)}$
• When events A and B are independent, the conditional probability simplifies to $P(A|B) = P(A)$, as the occurrence of event B does not affect the probability of event A (coin flips, dice rolls)

Bayes' Theorem

Bayes' theorem for probability updates

• Mathematical formula used to update probabilities when new information or evidence becomes available
• Expressed as $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$, where:
  • $P(A|B)$ represents the updated (posterior) probability of event A given new information B
  • $P(B|A)$ represents the likelihood of observing evidence B given that event A is true
  • $P(A)$ represents the prior probability of event A before considering evidence B
  • $P(B)$ represents the marginal probability of observing evidence B, calculated as $P(B) = P(B|A)P(A) + P(B|A^c)P(A^c)$, where $A^c$ denotes the complement of event A
• Particularly useful when there is prior knowledge or belief about the probability of an event (machine learning, spam filters)
• Incorporates new evidence or information that may affect the probability of the event (medical test results, customer feedback)

Engineering problems with Bayes' theorem

1. Identify the events of interest and the available evidence (system failures, product defects)
2. Determine the prior probabilities and likelihoods based on the problem statement or given data
3. Apply Bayes' theorem to calculate the updated (posterior) probabilities
4. Interpret the results in the context of the engineering problem

• Fault diagnosis in complex systems (aircraft engines, industrial machinery)
• Updating the probability of a product defect given inspection results (quality control, manufacturing)
• Estimating the likelihood of a particular cause given observed symptoms in a process (chemical plants, power generation)