"S", "S = {s: s is an outcome of the experiment}" "S E A", "s is an element of the set A" "S E/ A", "s is not an element of the set A" "A C B", "A is a subset of B" "A bar", "Complement of set A, {s:sE omega,sE/A}" "outcome", "An outcome is a possible result w of a random experiment" "sample space", "The sample space is the collection of all possible outcomes of a random experiment" "Venn diagrams", "Venn diagrams are pictorial representations of a set and its elements" "visualize Venn Diagrams", "" "Operations of Sets", "" "A random experiment", "A random experiment is an experiment or observation which can be performed (at least in theory) any number of times under the same relevant conditions" "Examples of a random experiment", "• Toss a fair coin<br /> • Roll a die<br /> • Measured times between arrivals in an intersection<br /> • Count the number of rainy days in a year" "Random Event", "" "Axioms of Probability", "" "Let A and B be events in omega<br /> A and B are independent if", "• P[A B] = P[A]P[B]<br /> • P[A|B] = P[A] and P[B|A] = P[B]" "Let A and B be events in omega<br /> If A and B are mutually exclusive, then", "• P[A U B] = P[A] + P[B] • P[A|B] = 0" "mutually independent", "A family of events A1, A2, ..., An is mutually independent if, for every subset of r different events Ai1, Ai2, ..., Air and for r = 2, 3, ..., n we have<br /> P[Ai1N Ai2N ...N Air ]<br /> = P[Ai1]P[Ai2] ... P[Air]" "Other basic properties in Axioms of Probability", "1. If A is a subset of B, then P[B-A] = P[B] – P[A]<br /> 2. P[Ac] = 1 – P[A]<br /> 3. P[A U B] = P[A] + P[B] – P[A B], and in general, for n events (A1, A2,..., An)" "", "" "Expected Cost of Failure with 2 sequential pumps", "Assume if the system fails, the opportunity cost and cost of fixing totals $600,000 per failure. What is the expected cost of failure?<br /> 0.0975(600,000)=$58,500" "", "" "Conditional Probability", "" "Law of Total Probability", ""
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