Ch 13 - Probability12th Mathematics
Definition of Conditional Probability
Definition of Conditional Probability
State the precise algebraic definition of the Conditional Probability of event E given that event F has already occurred.
Derivation
SOP (Intuition):
- The given event becomes the new, restricted sample space.
- We divide the favourable overlapping intersection area by the new total area to correctly re-scale the probability.
Component Anatomy:
- : Probability of event given that has officially occurred.
- : Probability of the simultaneous occurrence of both and .
- : Probability of the given event .
Boundary Conditions / Domain:
- Strict Constraint: Requires . You mathematically cannot condition on an impossible event.
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Conditional Probability of the Sample Space
What is the conditional probability of the entire sample space (or the event itself) given that the event has strictly occurred?
What is the conditional probability of the entire sample space (or the event itself) given that the event has strictly occurred?
.
- Reasoning: Once event is strictly known to have occurred, practically becomes the new certain universe. The probability that an outcome belongs to the sample space (or itself) given that it strictly belongs to is mathematically 100% ().
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Conditional Probability of a Complement
Conditional Probability of a Complement
State the fundamental formula relating the conditional probability of an event E and its complement E', strictly given F.
Component Anatomy:
- : The complement of event (i.e., the strictly defined event where does not occur).
- : The governing conditioning event.
Boundary Conditions / Domain:
- Constraint: .
- Conditional probability completely obeys standard probability axioms within the new restricted sample space .
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Also in this chapter
27 more cards to study in Prism
- 4Addition Theorem for Conditional Probability
- 5Distinction: $P(E|F)$ vs $P(F|E)$
- 6Multiplication Theorem (Two Events)
- 7Multiplication Theorem Chain Rule (Three Events)
- 8Constraint: With vs. Without Replacement
- 9Independent Events: Product Rule Test
- 10Trap: Independent vs. Mutually Exclusive
- 11Independence under Complementation
- 12Mutual Independence of Three Events
- 13Shortcut: Probability of 'At Least One'
- 14Definition: Partition of a Sample Space
- 15Theorem of Total Probability
- 16Bayesian Terminology: Prior, Likelihood, Posterior
- 17Bayes' Theorem Formula
- 18Odds Form of Bayes' Theorem
- 19Definition: Random Variable
- 20Constraints: Valid Probability Distribution
- 21Mean (Expectation) of a Random Variable
- 22Formula: Variance of a Random Variable
- 23Identification: Bernoulli Trials
- 24Binomial Distribution (Probability Mass Function)
- 25Mean and Variance of Binomial Distribution
- 26Subset Conditioning and Probability Bounds
- 27Symmetry of Conditional Probability Bounds
- 28Algebraic Analogue: Hidden Conditional Certainty
- 29Formula: Alternating Turns Game (Infinite G.P.)
- 30Application Trap: The $2\times 2$ Determinant
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