Sample Spaces & Events
A sample space
Set of all possible outcomes, e.g., {H,T}. Defines decision boundary of uncertainty.
P(rolling 3) if a Fair die
1/6; uniform chance across 6 outcomes.
P(A∪B)
Chance at least one occurs; overlap included.
Probability of B given A.
Narrows uncertainty to condition A.
Independent events
One’s occurrence doesn’t affect the other.
P(A)=0.25, interpret.
25% likelihood; Judgment: rare event; Decision: prepare for alternate outcomes.
Equally-likely may fail in practice?
Bias or unequal conditions; test fairness before modeling.
P(B|A1)=0.2 and P(B|A2)=0.5
Allocate monitoring or preventive resources toward A2 first.
P(B|A) = 0.4
B occurs in 40% of A cases.
P(A)=0.4,P(B)=0.5,P(A∩B)=0.2
Independent
Mutually exclusive vs exhaustive?
Exclusive: can’t co-occur. Exhaustive: cover all outcomes; ensures total probability =1.
Outcomes tossing two coins
4: HH, HT, TH, TT; forms full sample space.
P(A∩B)=0.1
10% co-occurrence; shared exposure; plan coordination.
P(B|A) vs P(A|B).
Reverses cause/effect; crucial distinction for reasoning.
P(A∩B)>P(A)P(B)
Positive correlation; events reinforce each other.
P(A∩B) if If A and B are disjoint
Zero; events cannot overlap.
Number of states if 3 machines each fail or not fail.
8 total; grows exponentially with components.
Complement A′
‘Not A’; represents failure or alternative case.
P(B|A)=1
B always follows A; deterministic link.
P(A∩B)<P(A)P(B)
Negative correlation; one reduces chance of the other.
If total probability ≠1
Model incomplete or overlapping; reassess definitions.
Enumerate outcomes
Ensures no missed or double-counted possibilities.
Reliability if A=success with P(A′)=0.2.
80%; fairly stable process but monitor 20% gap.
Evaluating probabilities conditionally for a process having different outcomes under varying conditions?
Finding likelihoods shift under specific situations, guiding targeted actions, policies, or resource allocation.
Assuming independence wrongly.
Underestimation of joint failures; policy blind spots.