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
• Roll a die
• Measured times between arrivals in an intersection
• Count the number of rainy days in a year
Random Event
Axioms of Probability
Let A and B be events in omega
A and B are independent if
• P[A B] = P[A]P[B]
• P[A|B] = P[A] and P[B|A] = P[B]
Let A and B be events in omega
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
P[Ai1N Ai2N ...N Air ]
= 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]
2. P[Ac] = 1 – P[A]
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?
0.0975(600,000)=$58,500
Conditional Probability
Law of Total Probability