Chapter 4: Probability and Counting Rules

Sample Space and Probability

General Terms

Probability Experiment: Random process with well defined potential results

Outcome: Result of a probability experiment

Event: Set of one or more outcomes

Simple event: Event which includes a single outcome, such as rolling a four
Compound event: Event which can be any of multiple outcomes, such as rolling odd

Tree diagram: Branched diagram showing all possible outcomes of a probability experiment

Sample Space: Set of all possible results of a probability experiment

Classical Probability

Classical Probability: Deals with events with equally likely sample space outcomes

Formula:

\[P(E)=\frac{n(E)}{n(S)}\]

where:

$P(E)$ is the probability of event $E$ occuring;

$n(E)$ is the number of outcomes included in event $E$; and

$n(S)$ is the total number of outcomes

Probability Rules:

Events have a chance between 0 and 1 to occur

0 is impossible. 1 is definite.

The total of all sample space events is 1

Complementary Events:

An event’s complementary event is an event’s negation; all outcomes not included in the event

Notation:

Event: $E$

Event complementary: $\overline{E}$

Formula:

\[E+\overline{E}=1\]

Empirical Probability

Empirical Probability: Deals with outcomes which do not have an equal chance of occurring, rather, each outcome has a relative frequency

The frequency table is constructed based on experiments

Formula:

\[P(E)=\frac{f(E)}{n}\]

where:

$P(E)$ is the probability of event $E$ occurring;

$f(E)$ is $E$’s relative frequency; and

$n$ is the sum of all frequencies

Subjective Probability

Subjective Probability uses a probability value based on an educated guess

The Addition Rules for Probability

Addition Rules

Addition Rules: Calculate the probability that either of multiple events will occur; “A or B”

Mutually Exclusive Events: Events that have no shared outcomes

Where two events are mutually exclusive, the probability for “A or B” to occur is the probability for A plus the probability for B

Formula:
\[⁍\]
Non mutually exclusive events:

Where two events are non mutually exclusive, the probability for “A or B” to occur is the probability for A, plus the probability for B, minus the probability of “A and B” occurring

Reasoning: events which overlap A and B were counted twice

Formula:
\[P(AB)=P(A)+P(B)-P(AB)\]

The Multiplication Rules and Conditional Probability

Multiplication Rules:

Multiplication Rules: Calculates the probability that all of multiple results will occur; “A and B”

Independent Events: Separate events in which one does not affect the outcome of the other

The probability of two independent events both occurring is the product of both events’ probability

Formula:

\[P(AB)=P(A)\cdot P(B)\]

Dependent Event: Events in which the outcome of one is affected by and dependent on the outcome of the other

Conditional Probability: Where event A affects the probability of event B, the probability of event B happening after event A happens is expressed as:

\[P(B|A)\]

Dependent events multiplication rules: The probability of two dependent events both occurring is the product of the both events’ probability:

Formula:

\[P(AB)=P(A) \cdot P(B|A)\]

Conditional Probability Rule: To specifically find the probability of event B occurring assuming event A has already occurred, divide P(AB) by P(A), since we’re assuming A already occurred, and are not accounting for A’s probability.

Formula:

\[P(B|A) = \frac{P(AB)}{P(A)}\]

Counting Rules

Fundamental Counting Rule

Fundamental Counting Rule: When multiple independent events have numerous possibilities, the number of total possibilities is the product of all the events’ possibilities

Example:

Event A has three possibilities, Event B has five, Event C has two.

$n_1 = 3;\space n_2=5; \space n_3=2$

$n_1\cdot n_2\cdot n_3= 30$

Permutation

Permutation: Specific sequence of a set number of items

Notation: The permutations of $n$ objects is expressed as $_nP$

Rule: $n$ objects have $n!$ total permutations

Formula:

\[_nP=n!\]

Where the sequence will only have $r$ elements, and not all $n$ elements: The permutations are expressed as $_nP_r$

To calculate: Divide the number of total $n$ permutations by the permutations of the difference between $r$ and $n$

Reasoning: each permutation with only $r$ elements is repeated again for each permutation of the subsequent elements

Formula:

\[_nP_r=\frac{n!}{(n-r)!}\]

Where there is $r$ number identical items: Divide $n!$ by $r!$

Reasoning: The specific sequence of the $r$ elements do not matter; the permutations were repeated for each individual $r!$ permutation.

Formula:

\[\frac{n!}{r!}\]

Example: If there are 12 total items, and 3 of them are identical, the total permutations is $\frac{12!}{3!}$

If there are multiple repeated values: Divide $n!$ by the product of the $r!$s

Reasoning: The total number of repeats is the product all the $r!$s; each individual $r!$ was repeated for each permutation of the other $r!$s

Formula:

\[_nP=\frac{n!}{r_1! \cdot r_2! \cdot r_3!...}\]

Example: If there are 12 total items, and 3 share one value, and 5 share another, the total permutations is $\frac{12!}{3!\cdot 5!}$

Combination

Combination: Combination of $r$ items out of a set of $n$ items

Example: In a set of 4 items, there are 6 possible combinations of 2 items.

Notation: The total number of combinations is expressed as $_nC_r$

Calculation: Divide $n!$ by the product of $r!$ and $(n-r)!$

Reasoning: In $n!$, each combination is repeated for each combination sequence, which is $r!$, and each of those are repeated for each sequence of the remaining items, which is $(n-r)!$

Formula:

\[_nC_r = \frac {n!} {r! \cdot (n-r)!}\]