4.2 Probability

Probability

Tree diagram

Union

De Morgan's Law

Independent

Events

Mutually exclusive

Addition law

Bayes Theorem

Conditional

Experimental

Theoretical

Cumulative frequency

There are two types of probability, experimental and theoretical.

Terminology	Definition
Number of trials	Total number of trials the experiment is repeated.
Outcomes	Different results possible for one trial of the experiment.
Frequency	Number of times of a certain outcome being observed.
Relative Frequency	Frequency represented as a fraction against the total number of trials.
Cumulative Frequency	The total of a frequency and all frequencies in a frequency distribution until a certain defined class interval.
Sample space $U$	Set of all possible outcomes of an experiment.

2D Grid

2 dimensional representation method of all outcomes.

Tree Diagram

Tree structure representation method of all outcomes.

Table of outcomes

Table representation method of all outcomes.

Intersection

The set of elements that exist in both

A

and

B

Union

The set of elements that exist in either

A

B

De Morgan’s Law	$(A \cap B)' = A'\cup B'$
Event $A$	A set of outcomes to which a probability is assigned. The probability of event $A$ is $P(A)$ .
Complementary Event $A'$	Two events are complementary if exactly one of them must occur. $P(A) + P(A') = 1$
Compound Event	A compound event denotes a situation where multiple events occur simultaneously or in succession.
Independent Events	A and B are independent iff $P(A\cap B) = P(A)P(B)$ or $P(A \| B) = P(A \| B') = P(A)$ This formula extends to more than two events.
Dependent Events	$A$ and $B$ are not independent; i.e. the occurrence of one event does affect the occurrence of the other event.
Mutually Exclusive Events	$A$ and $B$ are mutually exclusive if $A \cap B = ∅$ . $P(A \cup B) = P(A) + P(B)$
Sampling	The process of selecting one of the objects at random and inspecting it for particular features.
Addition Law	Two events: $P(A \cup B) = P(A) + P(B) - P(A\cap B)$ Three events: $P(A \cup B \cup C) = P(A) + P(B) + P(C) -P(A\cap B) - P(B \cap C) - P(C\cap A)+P(A\cap B\cap C)$ P(A \cup B \cup C) = P(A) + P(B) + P(C) -P(A\cap B) - P(B \cap C) - P(C\cap A)+P(A\cap B\cap C)
Conditional Probability	Probability of $A$ given $B$ is the probability of $A$ occurs given $B$ has occurred. $P(A \| B) = \frac{P(A\cap B)}{P(B)}$
Bayes Theorem	$P(A \| B) = \frac {P(B \| A)P(A)}{P(B)}$
Theoretical Probability	This is equivalent to the relative frequency.
Experimental Probability	$P(A) = \frac {n(A)}{n(U)}$