2023-09-25
Define basic terms related to events such as events, outcomes, and sample space.
Use proper set notation for events
Characterize possible outcomes, when something random occurs
Describe events into which outcomes can be grouped
Define important terms and rules within set theory such as unions, intersections, complements, mutually exclusive, and De Morgan’s Laws
Suppose you toss one coin.
What are the possible outcomes?
What is the sample space?
What are the possible events?
Suppose you toss one coin.
What are the possible outcomes?
Heads (\(H\))
Tails (\(T\))
Note
When something happens at random, such as a coin toss, there are several possible outcomes, and exactly one of the outcomes will occur.
Definition: Sample Space
The sample space \(S\) is the set of all outcomes
Definition: Event
An event is a collection of some outcomes. An event can include multiple outcomes or no outcomes.
What is the sample space?
What are the possible events?
When thinking about events, think about outcomes that you might be asking the probability of.
Suppose you toss two coins.
What is the sample space? Assume the coins are distinguishable
What are some possible events?
\(A =\) exactly one \(H =\)
\(B =\) at least one \(H =\)
Suppose you keep sampling people until you have someone with high blood pressure (BP)
What is the sample space?
Let \(H =\) denote someone with high BP.
Let \(H^C =\) denote someone with not high blood pressure, such as low or regular BP.
Definition: Union
The union of events \(A\) and \(B\), denoted by \(A \cup B\), contains all outcomes that are in \(A\) or \(B\) or both
Definition: Intersection
The intersection of events \(A\) and \(B\), denoted by \(A \cap B\), contains all outcomes that are both in \(A\) and \(B\).
Venn diagrams
Definition: Complement
The complement of event \(A\), denoted by \(A^C\) or \(A'\), contains all outcomes in the sample space \(S\) that are not in \(A\) .
Definition: Mutually Exclusive
Events \(A\) and \(B\) are mutually exclusive, or disjoint, if they have no outcomes in common. In this case \(A \cap B = \emptyset\), where \(\emptyset\) is the empty set.
Venn diagrams
Suppose you have \(n\) subjects in a study.
Let \(H_i\) be the event that person \(i\) has high BP, for \(i=1\ldots n\).
Use set theory notation to denote the following events:
Event subject \(i\) does not have high BP
Event all \(n\) subjects have high BP
Event at least one subject has high BP
Event all of them do not have high BP
Event at least one subject does not have high BP
Suppose you have \(n\) subjects in a study.
Let \(H_i\) be the event that person \(i\) has high BP, for \(i=1\ldots n\).
Use set theory notation to denote the following events:
Event subject \(i\) does not have high BP
Event all \(n\) subjects have high BP
Event at least one subject has high BP
Event all of them do not have high BP
Event at least one subject does not have high BP
Theorem: De Morgan’s 1st Law
For a collection of events (sets) \(A_1, A_2, A_3, \ldots\)
\[\bigcap\limits_{i=1}^{n}A_i^C = \Big(\bigcup\limits_{i=1}^{n}A_i\Big)^C\]
“all not A = \((\)at least one event A\()^C\)”
Theorem: De Morgan’s 2nd Law
For a collection of events (sets) \(A_1, A_2, A_3, \ldots\)
\[\bigcup\limits_{i=1}^{n}A_i^C = \Big(\bigcap\limits_{i=1}^{n}A_i\Big)^C\]
“at least one event not A = \((\)all A\()^C\)”
These laws also hold for infinite collections of events.
Draw Venn diagrams to convince yourself that these are true!
These laws are very useful when calculating probabilities.
This is because calculating the probability of the intersection of events is often much easier than the union of events.
This is not obvious right now, but we will see in the coming chapters why.
Chapter 1 Slides