Understanding Probability Basics

Probability helps us understand how likely events are to occur in uncertain situations.
It measures the chance of different outcomes and is crucial in fields like data science, statistics, and machine learning.

By understanding probability, we can analyze patterns, make predictions, and quantify uncertainty.

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: 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:

$P(\text{Head on a coin}) = \tfrac{1}{2}$
$P(\text{6 on a die}) = \tfrac{1}{6}$

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:

$P(\text{first card is an Ace}) = \tfrac{4}{52}$
$P(\text{second card is an Ace | first card was an Ace}) = \tfrac{3}{51}$

Then:

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

1. A coin is flipped twice. What is the probability of getting at least one head?

2. If $P(A) = 0.4$ , $P(B) = 0.6$ , and $P(A \cap B) = 0.2$ , what is $P(A \cup B)$ ?

3. In a deck of 52 cards, what is the probability of drawing a red king?

4. A die is rolled. If $A = \{1,3,5\}$ and $B = \{2,3,4,6\}$ , what is $P(A \cup B)$ ?

A coin is flipped twice. What is the probability of getting at least one head?

Select the correct answer

\frac{1}{4}

\frac{1}{2}

\frac{3}{4}

1

If $P(A) = 0.4$ , $P(B) = 0.6$ , and $P(A \cap B) = 0.2$ , what is $P(A \cup B)$ ?

Select the correct answer

1.0

0.8

1.2

0.6

In a deck of 52 cards, what is the probability of drawing a red king?

Select the correct answer

\frac{1}{26}

\frac{2}{52}

\frac{1}{13}

\frac{4}{52}

A die is rolled. If $A = \{1,3,5\}$ and $B = \{2,3,4,6\}$ , what is $P(A \cup B)$ ?

Select the correct answer

\frac{6}{6}

\frac{1}{6}

\frac{4}{6}

\frac{3}{6}

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Understanding Probability Basics

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By understanding probability, we can analyze patterns, make predictions, and quantify uncertainty.

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: 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