CH 2.1 Probability Basic Ideas Flashcards
Define Sample Space
The set of all possible outcomes of an experiment.
Define Event
A subset of a sample space.
Define Union of Two Events
The set of outcomes that belong to either to A, B, or both.
Define Intersection of Two Events
Is the set of outcomes that belong to both A and B.
Define Complement of an Event
Is the set of outcomes that do not belong the corresponding event.
Define Events that are Mutually Exclusive
Events that don’t have any outcomes in common.
Define Probability
Given an experiment and any event A: The expression P(A) denotes the probability that the event A occurs. P(A) is the proportion of times that event A would occur in the long run, if the experiment were to repeated over and over again.
Define the 3 Axioms of Probability
- Let S be a sample space. Then P(S) = 1.
- For any event A, 0</= P(A) </= 1
- If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B).
Formula for the probability of the complement of an event.
P(A^c) = 1 - P(A).
Formula for the probability of an empty set.
P(∅) = 0.
Formula for the probability of an event in a sample space.
P(A) = k/N.
+ N is the equally likely outcomes of a sample space.
+ k is the outcomes of an event in the sample space.
Formula for the union of two events that are mutually exclusive.
P(A or B) = P(A) + P(B) - P(A and B)
Write Out the Proof P(A^c) = 1 - P(A)
Let S be a sample space and let A be an event. Then A and A^c are mutually exclusive, so by Axiom 3,
P(A ∪ A^c) = P(A) + P(A^c)
But A ∪ A^c = S, and by Axiom 1, P(S) = 1.
Therefore, P(A ∪ A^c) = P(S) = 1.
It follows that P(A) + P(A^c) = 1,
so P(A^c) = 1 − P(A).
Write out proof that the probability of an empty sample space is equal to zero.
Let S be a sample space.
Then ∅ = S^c. Therefore P(∅) = 1 − P(S) = 1 − 1 = 0.
