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