Review
• Law of Large Numbers
As the number of trials increases, the proportion of
occurrences of any given outcome approaches a particular
number “in the long run”.
(We need to assume that the outcome of any one trial does
not depend on the outcome of any other trial.)
• Probability
With a randomized experiment or a random sample or other
random phenomenon (such as a simulation), the probability
of a particular outcome is the proportion of times that the
outcome would occur in a long run of observations.
• Independent Trials
Different trials of a random phenomenon are independent if
the outcome of any one trial is not affected by the outcome of
any other trial.
How to Find Probabilities?
• In practice, the sample proportion estimates the actual
probability.
• Sample Space
For a random phenomenon, the sample space is the set of all
possible outcomes.
• Event
An event is a subset of the sample space.
Probabilities for a Sample Space
• The probability of each individual outcome is between 0 and 1.
• The total of all the indicidual probabilities equals 1.
Probability of an Event
• The probability of an event A, denoted by P(A), is obtained
by adding the probabilities of the individual outcomes in the
event.
• When all the possible outcomes are equally likely,
P(A) =
number of outcomes in event A
number of outcomes in the sample space
In practice, equally likely outcomes are unusual.
• Complement of an Event
The complement of an event A consists of all outcomes in
the sample space that are not in A. It is denoted by Ac .
The probabilities of A and of Ac add to 1, so
P(Ac ) = 1 − P(A)
It’s sometimes easier to find the probability of the
complement than to find the probability of the event itself.
• Disjoint Event
Two events, A and B, are disjoint if they do not have any
common outcomes.
• Intersection of Two Event
The intersection of A and B consists of outcomes that are in
both A and B.
• Union of Two Event
The union of A and B consists of outcomes that are in A or
B. In probability, “A or B” denotes that A occurs or B occurs
or both occur.
• Addition Rule
For the union of two events,
P(A or B) = P(A) + P(B) − P(A and B).
If the events are disjoint, then P(A and B) = 0, so
P(A or B) = P(A) + P(B).