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Chance behavior is unpredictable in the short run but has a regular and predictable
pattern in the long run.
We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a
regular distribution of outcomes in a large number of repetitions.
The probability of any outcome of a random phenomenon is the proportion of times the outcome
would occur in a very long series of repetitions. That is, probability is a long-term relative
The sample space S of a random phenomenon is the set of all possible outcomes.
The event is any outcome or a set of outcomes of a random phenomenon. That is, an event is a
subset of the sample space.
A probability model is a mathematical description of a random phenomenon consisting of two
parts: a sample space S and a way of assigning probabilities to events.
1. The probability P(A) of any event satisfies 0 P(A)  1.
2. If S is the sample space in a probability model, then P(S) = 1.
3. The complement of any event A is the event that A does not occur, written as Ac or A’.
The complement rule states that
P(Ac) = 1 – P(A)
P(Ac) = P(A’)
4. The union of any collection of events is the event that at least one of the collection occurs
so 1 or the other or both.
General Rule for any two events A and B,
P(A or B) = P(A) + P(B) - P(A and B)
P(A  B) = P(A) + P(B) - P(A B)
5. Two events A and B are disjoint (also called mutually exclusive) if they have no outcomes in
common so can never occur simultaneously. If A and B are disjoint,
P(A or B) = P(A) + P(B)
This is the addition rule for disjoint events.
6. The intersection of any collection is the event that ALL of the events occur. This comes
from the conditional probability formula
P(B/A) = P(A  B)
so P(A and B) = P(A)P(B given A)
This is the multiplication rule.
P(A  B) = P(A) P(B/A)
Two events A and B are independent if knowing that one occurs does not change the probability
that the other occurs.
P(B/A)=P(B) So, if A and B are independent then P(A and B) = P(A)P(B)
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