Chapter 5: Probability Flashcards
PROBABILITY
The study of understanding and determining the outcomes of events yet to happen.
A measure of the likelihood of chance behavior.
Along-term prognosis.
EXPERIMENT
In probability, any process with uncertain results that can be repeated.
OUTCOME
One of the results of an experiment
SAMPLE SPACE (“S”)
The collection of all outcomes from a probability experiment
EVENT (“E”, “F”, etc.)
A subset of the sample space
Intersection of 2 Events E and F
The collection of all outcomes in both E and F. Denoted E and F or E ∩ F):
Disjoint (mutually exclusive) Events
Events are disjoint if they have NO OUTCOMES IN COMMON.
Union of 2 Events E and F
The collection of all outcomes in either E or F or both. Denoted E or F or E ∪ F):
Complement of an Event E
The collection of outcomes in the sample space NOT IN E. Denoted E^C
Impossible Event
An event E such that the probability of E = 0.
Sure Event
An event E such that the probability of E = 1.
Unusual Event
Event: An event E such that the probability of E ≤ 0.05.
THE FUNDAMENTAL RULES OF PROBABILITY
- The probability of any event E, P(E), is a NUMBER that must be greater than or equal to 0 and less than or equal to 1. That is, 0 ≤ P(E) ≤ 1
- If the sample space S = {e1, e2, …, en}, then P(e1) + P(e2) + … + P(en) = 1. That is, the sum of the probabilities of all the outcomes must equal 1. Another way to write this is that P(S) = 1.
Classical Method of Computing Probabilities
Assumes equally likely outcomes; that is, all Outcomes are assumed to have the same probability of occurring.
So, if S is the sample space of this experiment,
P(E) = N(E)/N(S)
where N(E) is the number of outcomes in E, and N(S) is the number of outcomes in the sample space.
Marginal Probability (Unconditional Probability)
When an experiment involves more than one event, the probability of one of the events, irrespective of the presence of the other events. It is computed as the relative frequency of the event relative to the total population.
Joint Probability of Two Events, E and F
The probability of the intersection of two events; namely P(E and F).
Conditional Probability of Two Events
If E and F are any two events, then:
P(F / E) = ( 𝑷(𝑭 𝒂𝒏𝒅 𝑬))/(𝑷(𝑬)) = ( 𝑷(𝑬 𝒂𝒏𝒅 𝑭))/(𝑷(𝑬))
The expression P(F / E) is read “the probability of event F given event E”.
General Multiplication Rule
The probability that two events E and F both occur is:
P(E and F) = P(E) * P(F|E)
Independent Events, E and F
The occurrence of event E in a probability experiment does not affect the probability of event F (or vice versa).
Computing Probabilities Using the Relative Frequency (Empirical) Method
For an event E of an experiment, we define the relative frequency of E as
( the number of times the event E was observed)/(the number of repetitions of the experiment)
We then define the P(E) to equal this relative frequency
Computing Probabilities Using the Relative Frequency (Empirical) Method
For an event E of an experiment, we define the relative frequency of E as
( the number of times the event E was observed)/(the number of repetitions of the experiment)
We then define the P(E) to equal this relative frequency
General Addition Rule for Joint Events
If E and F are disjoint (mutually exclusive) events, then
P(E pr F) = P(E) + P(F)
Probability Model
A list of the possible outcomes of an experiment and each outcome’s probability. Must satisfy both fundamental rules of probabilities
Probability Model
A list of the possible outcomes of an experiment and each outcome’s probability. Must satisfy both fundamental rules of probabilities, (0 ≤ P(E) ≤ 1) and
P(S) = 1.
The Empty Event
p(E) = 0
The Law of Large Numbers
As the number of repetitions of a probability experiment increases, the proportion (relative frequency) with which a certain outcome is observed gets closer to the classical probability of the outcome.
Complement Rule
If E represents any event and E^C represents the complement of E, then: P(E) + P(E^C) = 1 - P(E^C)
Multiplication Rule for Independent Events
If E and F are independent events, then
P(E and F) = P(E) * P(F)
Simulation
A technique used to recreate a random event
Contingency Table (Two-Way Table)
A table that relates two categories of data.
Dependent Events, E and F
The occurrence of event E in a probability experiment affects the probability of event F (or vice versa).
