Chapter 6: Probability Flashcards
Random experiment
An action or process that leads to one of several possible outcomes
- need exhaustive list of mutually exclusive outcomes
Sample space
The sample space of a random experiment is a list of all possible outcomes of that experiment.
List must be exhaustive and mutually exclusive
Use set notation to represent all outcomes
S={. }
Requirements of probabilities
Given a sample space the probabilities assigned to outcomes must satisfy two requirements
Where P(Oi) is the probability of outcome i
1) probability of any outcome must lie between 0 and 1
0>= P(Oi) <=1
2) the sum of the probabilities of all the outcomes in an sample space must be 1
Classical approach of assigning probabilities
When answer is equally possible/ pure chance the probability of each outcome is 1/Number of outcomes
Relative frequency approach for assigning probabilities
Probability= long run frequency with which an outcome occurs
Samples to determine relative probabilities (history of outcomes necessary)
Subjective approach to assigning probabilities
Used when classical approach isn’t reasonable and there is no history of outcomes to study
Probability = degree of belief we hold in the occurrence of an event (potential from the study of related factors
Simple event
An individual outcome of a sample space
Event
A collection or set of one or more simple events in a sample space
Probability of an event
The sum of the probabilities of the simple events that constitute the event
Intersection of events A and B
The event that occurs when both events A and B occur. Denoted as: A and B
Probability of intersection = joint probability
Joint probability
Probability that a specific intersection of events will occur
Joint probability of all possible combinations = 1
Marginal probabilities
Calculated from joint probability tables (in the margins)
Sum of probability of each district event
Conditional probability
Probability of one event given the occurrence of another, related, event
P(B1| A1) (probability of B1 given A1)
P(A|B) = P(A and B) / P(B)
Probability of both events occuring divided by the probability of the given event occuring
Aka ratio of joint probability to marginal probability
Independent events
Events are independent if the probability of one event is not affected by the occurances of the other event
If P(A|B) = P(A)
If probability not equal then the events are dependant
Union of events
A union of events occurs with either event or both events occur. Denoted as:
A or B
Sum of all probabilities of possible outcomes: A only, B only, A and B
(or 1 minus the probability that neither event occurs)
Complement of an event
The event that occurs when event A does not occur (Ac)
Complement Rule
Probability of an event and it’s complement must sum to 1
For any event A
Probability of the complement of A = 1 - probability of A
P(Ac) = 1 - P(A)
Multiplication rule
Used to calculate the joint probability of two events
P(A and B) = P(B) * P(A|B)
- for dependent events
(Also = P(A) * P(B|A)
For independent events:
P(A and B) = P (B) * P(A)
Addition rule
Calculated the probability of the union of two events
(Probability that event A, or event B, or both occur)
P (A or B) = P(A) + P(B) - P (A and B)
(Because the probability of A and B is already included in P(A)+P(B))
Addition rule for mutually exclusive evets
P (A and B) = P(A) + P(B)
Probability tree
Events in an experiment represented by lines that branch at each choice/instance
Total probability of all ending joint probabilities should be 1
For mutually exclusive events probability trees are added not multipled
Without replacement
Options from a limited pool are not replaced after first choice is made, changing the probability of the next pick
(Vs with replacement where probability of a given pick is the same every time because option is replaced)
Prior probabilities
Probabilities determined prior to the decision in question (when asking a question if x then what will happen to y these are the probabilities of Y if not X)
Likelihood probabilities
Conditional probabilities based on a possible decision
(when asking a question if x then what will happen to y these are the varies probilities of y if decision x is made or not made)
Posterior probabilities
Or revised probabilities
(when asking a question if x then what will happen to y, given X or not x)
Revised to reflect answer to decision
Bayes law
P(Ai|B) = (P(Ai)P(B|Ai))/ (P(A1)P(B|A1))+P(A2)P(B|A2))+…+(P(An)P(B|An))
Aka
Probability of Ai and B / probability of B
