Statistics Chapter 3
Venn Diagram
A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. Complements, intersections, and unions can be shown using Venn Diagrams.
Continuous Sample Space
A sample space is continuous if it contains an interval (either finite or infinite) of real numbers.
Discrete Sample Space
A sample space is discrete if it consists of a finite or countable infinite set of outcomes. Discrete sample space can be shown in list form, set notation, Venn diagram, and tree diagram.
Define a Trial
A single attempt of a random experiment is known as a trial. If we measure the electrical current in a thin copper wire, each measurement is a trial.
Contingency Table
A table that shows the distribution of one variable in rows and another in columns is known as contingency table.
Tree Diagram
A tree diagram use branches to show the different outcomes of experiments and makes complex probability questions easy to visualize.
Define an Event
An event is a specific collection of sample points. In other words, an event is a subset of the sample space of a random experiment. It is typically denoted with a capital letter. It can be a single outcome, several outcomes, entire sample space, S or null set i.e. no outcome at all (denoted by ∅).
Define an Outcome
An outcome of a trial is the value measured or observed. Such as if you flip a coin, the two possible outcomes are head or tail.
Define Complement of an Event
Complement of an event A (denoted by Ac) is the subset of all the elements of S (sample space) that are not in A. Such as, rolling a 5 or greater and rolling a 4 or less on a die are complementary events.
Define an Element / Member / Sample Point
Each outcome in the sample space is known as sample point. A sample point contains exactly one outcome. In other words, a sample point is the most basic outcome of an experiment.
Define Equally Likely Events
Events that have the same probability of occurring are called equally likely events. For example, if you toss a fair six-sided die, each face (1, 2, 3, 4, 5, 6) is equally likely to occurs in the sample space. If you randomly guess the answer to true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.
Define Compound Events
Events which are composed of two or more other events are called compound events. Compound events can be formed in one of the two ways: Intersection and Union.
Chance Experiment
If the results of an experiment are not predetermined, then the experiment is said to be a chance experiment. Such as, counting the number of thunderstorms on any given day of July.
Define Intersection (AND Event)
Intersection of A and B i.e. (A∩B) is defined as the event containing all of the elements that are common to both A and B.
Define an Experiment
It is defined as a planned operation carried out under controlled conditions. Such as, flipping one fair coin twice.
Define Probability
Probability is the proportion of times that an outcome occurs in a very large number of trials. Probability is used to quantify uncertainty and can be expressed as fraction, decimal or percentage. Probabilities are between zero and one, inclusive.
Marginal Probability
Probability of any single event occurring unconditioned on any other events is known as marginal probability. This type of probability depends only on totals found in the margins of the contingency table.
Joint Probability
Probability of two events occurring together is known as joint probability.
Define Conditional Probability
Suppose we conduct an experiment, and we know the event B has occurred, the probability of event A occurring given that B has already occurred is known as conditional probability. In general, we call this a conditional probability because the probability of the event A is conditional on the event B.
Define a Sample Space
The set of all possible outcomes of an experiment is known as sample space. We denote the sample space by S. In other words, the sample space of an experiment is the collection of all its sample points. Sample space can be discrete or continuous.
Define Independent Events
Two events A and B are independent if the outcome of one does not affect the outcome of other. Note: When the sampling is done with replacement, each member of a population is replaced after it is picked, then the member has the probability of chosen more than once. When the sampling is done this way, the events are considered to be independent.
Define Mutually Exclusive (Disjoint) Events
Two events A and B are said to be mutually exclusive (disjoint) if they can NOT occur at the same time. In other words, A and B are mutually exclusive if they have no elements in common. In this case P(A∩B) = 0.
Define Non-mutually Exclusive Events
Two events are said to be non-mutually exclusive events if they can occur at the same time. In this case P(A∩B) ≠ 0. Before learning the examples of mutually exclusive and non-mutually exclusive events, lets familiar with suites of playing cards.
Define Union (OR Event)
Union of A and B i.e. (A∪B) is defined as the event containing all of the elements that belong to either A or B or both.
Define Dependent Events
When the outcome of the first event affects the outcome of the second event, the two events are said to be dependent events. Note: When the sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent.
Random Experiment
a.An experiment is random if we know what outcomes could happen, but not which particular values will happen in any given trial. Such as, when we flip a fair coin, we know that we will get either a head or a tail.
