Events in probability:

• Events refer to the outcomes or sets of outcomes from an experiment.

Definition of an event:

• An event can be a single outcome or a combination of multiple outcomes.

Understanding sample space:

• Sample space is the set of all possible outcomes of an experiment.

Types of events:

• Simple event: An event consisting of a single outcome.
• Compound event: An event consisting of multiple outcomes.
• Equally likely event: When each outcome in the sample space has the same probability.

Concept of occurrence and non-occurrence:

• Occurrence: When an event takes place.
• Non-occurrence: When an event does not take place.

Definition of sample space:

• Sample space is the set of all possible outcomes of an experiment.

Examples of sample spaces:

• Sample space of flipping a coin: {Heads, Tails}
• Sample space of rolling a dice: {1, 2, 3, 4, 5, 6}

Cardinality of a sample space:

• Cardinality refers to the number of elements in a set.
• The cardinality of a sample space is the total number of outcomes.

Properties of sample space:

• The sample space is exhaustive, meaning it includes all possible outcomes.
• Each outcome in the sample space is unique and distinct.

Representing sample space using Venn diagrams:

• Venn diagrams can visually represent the sample space and its subsets.
• Each outcome is represented by a circle or set, and intersections represent common outcomes.

Definition of probability:

• Probability is a measure of the likelihood of an event occurring.

Understanding the concept of likelihood:

• Likelihood refers to the chance or probability of an event happening.

Probability of an event (P(A)):

• P(A) represents the probability of event A occurring.
• Probability values range from 0 to 1, including both endpoints.

Range of probability values:

• 0 ≤ P(A) ≤ 1

Certain and impossible events:

• A certain event has a probability of 1, meaning it will always occur.
• An impossible event has a probability of 0, meaning it will never occur.

Addition theorem in probability:

• The addition theorem deals with the probability of the union or intersection of two or more events.

Definition of mutually exclusive events:

• Mutually exclusive events cannot occur simultaneously.

Addition rule for mutually exclusive events:

• P(A or B) = P(A) + P(B)

Examples of mutually exclusive events:

• Getting a head or a tail in a coin toss.
• Rolling a 1 or a 3 on a fair six-sided die.

Addition rule for non-mutually exclusive events:

• P(A or B) = P(A) + P(B) - P(A and B)

Introduction to conditional probability:

• Conditional probability measures the likelihood of an event occurring given that another event has already occurred.

Definition of conditional probability (P(A|B)):

• P(A|B) represents the probability of event A occurring given that event B has already occurred.

Calculation of conditional probability:

• P(A|B) = P(A and B) / P(B)

Understanding the concept of independence:

• Independence refers to two events that do not affect each other’s occurrence.

Calculation of probability of independent events:

• P(A and B) = P(A) * P(B)

Multiplication theorem in probability:

• The multiplication theorem deals with the probability of the intersection of two or more independent events.

Definition of independent events:

• Independent events do not affect each other’s occurrence.

Multiplication rule for independent events:

• P(A and B) = P(A) * P(B)

Examples of independent events:

• Getting heads on a coin toss and rolling a 2 on a fair six-sided die.

Dependence between events:

• Events that are not independent are dependent on each other’s occurrence.

Introduction to Bayes’ theorem:

• Bayes’ theorem is used to calculate the conditional probability of an event given the occurrence of another event.

Use of Bayes’ theorem in conditional probability:

• Bayes’ theorem incorporates prior probabilities and new evidence to update the probability of the event of interest.

Formula for calculating Bayes’ theorem:

• P(A|B) = (P(B|A) * P(A)) / P(B)

Application of Bayes’ theorem in real-life scenarios:

• Medical diagnosis, spam filtering, weather forecasting, etc.

Understanding the concept of prior and posterior probabilities:

• Prior probability refers to the initial probability of an event.
• Posterior probability is the updated probability after considering new evidence.

Definition of complementary events:

• Complementary events are events that do not occur together.

Properties of complementary events:

• The probability of an event and its complementary event add up to 1.

Calculation of probabilities using complementary events:

• P(A’) = 1 - P(A)

Use of complement rule in probability problems:

• The complement rule can be used to find the probability of an event by subtracting its complement from 1.

Examples of complementary events:

• Getting a head or a tail in a coin toss.

Introducing permutations in probability:

• Permutations involve the arrangements of objects in a specific order.

Definition of permutations:

• Permutations consider the order of selection and repetition is not allowed.

Calculation of permutations:

• The number of permutations of n objects taken r at a time is denoted by nPr.
• The formula is nPr = n! / (n-r)!

Understanding the concept of ordered arrangements:

• The arrangement of objects matters in permutations.

Examples of permutations in different scenarios:

• Arranging students in a line, selecting a president, arranging letters in a word.

Introducing combinations in probability:

• Combinations involve the selections of objects without considering the order.

Definition of combinations:

• Combinations consider the groupings and repetition is not allowed.

Calculation of combinations:

• The number of combinations of n objects taken r at a time is denoted by nCr.
• The formula is nCr = n! / (r! * (n-r)!)

Understanding the concept of unordered selections:

• The arrangement of objects does not matter in combinations.

Examples of combinations in different scenarios:

• Selecting a committee, choosing lottery numbers, forming a poker hand.