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Definition

Probability is the measure of the likelihood that an event will occur. It quantifies uncertainty and is essential in fields like data science, statistics, and machine learning, helping us analyze patterns, make predictions, and assess risks.

The Basic Definition of Probability

The probability of an event $A$ occurring is given by:

P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

This formula tells us how many ways our desired event can happen compared to all possible outcomes. Probability always ranges from 0 (impossible) to 1 (certain).

Understanding Sample Space and Events

Sample space - all possible outcomes of an experiment;
Event - a specific outcome or set of outcomes we're interested in.

Example with flipping a coin:

Sample space = {Heads, Tails} ;
Event A = {Heads} .

Then:

P(A) = \frac{P(\text{Heads})}{P(\text{Heads}) + P(\text{Tails})} = \frac{0.5}{0.5+0.5} = 0.5

Union Rule: "A OR B Happens"

Definition: the union of two events $A \cup B$ represents outcomes where either $A$ occurs, or $B$ occurs, or both occur.

Formula:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

We subtract the intersection to avoid double-counting outcomes that appear in both events.

Union Example: Rolling a Die

Let's roll a six-sided die:

Event A = {1, 2, 3} (rolling a small number)
Event B = {2, 4, 6} (rolling an even number)

Union and intersection:

$A \cup B = \{1, 2, 3, 4, 6\}$
$A \cap B = \{2\}$

Calculations step-by-step:

P(A) = \frac{3}{6} = \frac{1}{2} \\[6pt] P(B) = \frac{3}{6} = \frac{1}{2} \\[6pt] P(A \cap B) = \frac{1}{6}

Apply the union formula:

P(A \cup B) = \frac{3}{6} + \frac{3}{6} - \frac{1}{6} = \frac{5}{6}

Intersection Rule: "A AND B Both Happen"

Definition: The intersection of two events $A \cap B$ represents outcomes where both $A$ and $B$ occur simultaneously.

General Formula

In all cases:

P(A \cap B) = P(A) \times P(B|A)

where $P(B|A)$ is the conditional probability of $B$ given that $A$ has already occurred.

Case 1: Independent Events

If the events do not affect each other (e.g., flipping a coin and rolling a die):

P(A \cap B) = P(A) \times P(B)

Example:

$P(\text{Head on a coin}) = \frac{\raisebox{1pt}{$1$}}{\raisebox{-1pt}{$2$}}$ ;
$P(\text{6 on a die}) = \frac{\raisebox{1pt}{$1$}}{\raisebox{-1pt}{$6$}}$ .

Then:

P(A \cap B) = \tfrac{1}{2} \times \tfrac{1}{6} = \tfrac{1}{12}

Case 2: Dependent Events

If the result of the first event influences the second (e.g., drawing cards without replacement):

P(A \cap B) = P(A) \times P(B|A)

Example:

$P(\text{first card is an Ace}) = \tfrac{\raisebox{1pt}{$4$}}{52}$ ;
$P(\text{second card is an Ace | first card was an Ace}) = \tfrac{\raisebox{1pt}{$3$}}{\raisebox{-1pt}{$51$}}$ .

Then:

P(A \cap B) = \tfrac{4}{52} \times \tfrac{3}{51} = \tfrac{1}{221}

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Section 5. Chapter 1

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Definition

The Basic Definition of Probability

The probability of an event $A$ occurring is given by:

P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

This formula tells us how many ways our desired event can happen compared to all possible outcomes. Probability always ranges from 0 (impossible) to 1 (certain).

Understanding Sample Space and Events

Sample space - all possible outcomes of an experiment;
Event - a specific outcome or set of outcomes we're interested in.

Example with flipping a coin:

Sample space = {Heads, Tails} ;
Event A = {Heads} .

Then:

P(A) = \frac{P(\text{Heads})}{P(\text{Heads}) + P(\text{Tails})} = \frac{0.5}{0.5+0.5} = 0.5

Union Rule: "A OR B Happens"

Definition: the union of two events $A \cup B$ represents outcomes where either $A$ occurs, or $B$ occurs, or both occur.

Formula:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

We subtract the intersection to avoid double-counting outcomes that appear in both events.

Union Example: Rolling a Die

Let's roll a six-sided die:

Event A = {1, 2, 3} (rolling a small number)
Event B = {2, 4, 6} (rolling an even number)

Union and intersection:

$A \cup B = \{1, 2, 3, 4, 6\}$
$A \cap B = \{2\}$

Calculations step-by-step:

P(A) = \frac{3}{6} = \frac{1}{2} \\[6pt] P(B) = \frac{3}{6} = \frac{1}{2} \\[6pt] P(A \cap B) = \frac{1}{6}

Apply the union formula:

P(A \cup B) = \frac{3}{6} + \frac{3}{6} - \frac{1}{6} = \frac{5}{6}

Intersection Rule: "A AND B Both Happen"

Definition: The intersection of two events $A \cap B$ represents outcomes where both $A$ and $B$ occur simultaneously.

General Formula

In all cases:

P(A \cap B) = P(A) \times P(B|A)

where $P(B|A)$ is the conditional probability of $B$ given that $A$ has already occurred.

Case 1: Independent Events

If the events do not affect each other (e.g., flipping a coin and rolling a die):

P(A \cap B) = P(A) \times P(B)

Example:

$P(\text{Head on a coin}) = \frac{\raisebox{1pt}{$1$}}{\raisebox{-1pt}{$2$}}$ ;
$P(\text{6 on a die}) = \frac{\raisebox{1pt}{$1$}}{\raisebox{-1pt}{$6$}}$ .

Then:

P(A \cap B) = \tfrac{1}{2} \times \tfrac{1}{6} = \tfrac{1}{12}

Case 2: Dependent Events

If the result of the first event influences the second (e.g., drawing cards without replacement):

P(A \cap B) = P(A) \times P(B|A)

Example:

$P(\text{first card is an Ace}) = \tfrac{\raisebox{1pt}{$4$}}{52}$ ;
$P(\text{second card is an Ace | first card was an Ace}) = \tfrac{\raisebox{1pt}{$3$}}{\raisebox{-1pt}{$51$}}$ .

Then:

P(A \cap B) = \tfrac{4}{52} \times \tfrac{3}{51} = \tfrac{1}{221}

Everything was clear?

Thanks for your feedback!

Section 5. Chapter 1