Probability - Random Experiment

  • Definition of probability
  • Random experiment and its outcomes
  • Sample space and events
  • Examples of random experiments
  • Types of events (simple, compound, complementary) Probability - Probability of an Event
  • Probability of an event
  • Empirical probability
  • Theoretical probability
  • Classical probability
  • Law of large numbers Probability - Addition Rule
  • Addition rule for probability
  • Mutually exclusive events
  • Addition rule for mutually exclusive events
  • Non-mutually exclusive events
  • Addition rule for non-mutually exclusive events Probability - Multiplication Rule
  • Multiplication rule for probability
  • Independent events
  • Multiplication rule for independent events
  • Dependent events
  • Multiplication rule for dependent events Probability - Conditional Probability
  • Conditional probability
  • Conditional probability formula
  • Independent and dependent events in terms of conditional probability
  • Multiplication rule for conditional probability
  • Bayes’ theorem Probability - Combinations
  • Combinations
  • Permutations
  • Combination formula
  • Example of combinations
  • Application of combinations in real-life scenarios Probability - Permutations
  • Permutations
  • Permutation formula
  • Example of permutations
  • Application of permutations in real-life scenarios Probability - Binomial Distribution
  • Binomial distribution
  • Bernoulli trials
  • Binomial probability formula
  • Example of binomial distribution
  • Properties of binomial distribution Probability - Normal Distribution
  • Normal distribution
  • Characteristics of normal distribution
  • Standard normal distribution
  • Z-score and its calculation
  • Examples of normal distribution

Probability – Random Experiment

  • Definition of probability: Probability is a measure of the likelihood of an event occurring in a random experiment.
  • Random experiment and its outcomes: A random experiment is an experiment in which the outcome cannot be predicted with certainty. The possible outcomes of a random experiment are called its sample space.
  • Sample space and events: The sample space of a random experiment is the set of all possible outcomes. An event is a subset of the sample space.
  • Examples of random experiments: Tossing a coin, rolling a dice, drawing a card from a deck, etc.
  • Types of events (simple, compound, complementary): A simple event consists of a single outcome. A compound event consists of more than one outcome. The complementary event of an event A is the event consisting of all outcomes that are not in A.

Probability – Probability of an Event

  • Probability of an event: The probability of an event is a number between 0 and 1 that represents the likelihood of that event occurring.
  • Empirical probability: Also known as experimental probability, it is calculated by conducting experiments and observing the outcomes.
  • Theoretical probability: It is calculated based on the number of favorable outcomes and the total number of possible outcomes.
  • Classical probability: It is used when all outcomes in the sample space are equally likely.
  • Law of large numbers: As the number of trials in a random experiment increases, the experimental probability approaches the theoretical probability.

Probability – Addition Rule

  • Addition rule for probability: The probability of the union of two events A and B is given by P(A∪B) = P(A) + P(B) - P(A∩B).
  • Mutually exclusive events: Two events are said to be mutually exclusive if they cannot occur at the same time.
  • Addition rule for mutually exclusive events: For mutually exclusive events A and B, P(A∪B) = P(A) + P(B).
  • Non-mutually exclusive events: Two events are non-mutually exclusive if they can occur at the same time.
  • Addition rule for non-mutually exclusive events: For non-mutually exclusive events A and B, P(A∪B) = P(A) + P(B) - P(A∩B).

Probability – Multiplication Rule

  • Multiplication rule for probability: The probability of the intersection of two events A and B is given by P(A∩B) = P(A) * P(B|A).
  • Independent events: Two events are independent if the occurrence of one event does not affect the occurrence of the other event.
  • Multiplication rule for independent events: For independent events A and B, P(A∩B) = P(A) * P(B).
  • Dependent events: Two events are dependent if the occurrence of one event affects the occurrence of the other event.
  • Multiplication rule for dependent events: For dependent events A and B, P(A∩B) = P(A) * P(B|A).

Probability - Conditional Probability

  • Conditional probability: The probability of an event B occurring given that event A has already occurred is called conditional probability and is denoted by P(B|A).
  • Conditional probability formula: P(B|A) = P(A∩B) / P(A), where P(A) ≠ 0.
  • Independent and dependent events in terms of conditional probability: Two events A and B are independent if and only if P(B|A) = P(B).
  • Multiplication rule for conditional probability: For events A and B, P(A∩B) = P(A) * P(B|A).
  • Bayes’ theorem: It states that for any two events A and B, P(A|B) = [P(B|A) * P(A)] / P(B), where P(B) ≠ 0.

Probability - Combinations

  • Combinations: A combination is a selection of objects without regard to the order. The number of combinations of r objects from a set of n objects is given by the formula: C(n, r) = n! / (r!(n-r)!), where n ≥ r.
  • Permutations: A permutation is an arrangement of objects in a certain order. The number of permutations of r objects from a set of n objects is given by the formula: P(n, r) = n! / (n-r)!, where n ≥ r.
  • Combination formula: The combination formula is used to calculate the number of ways to choose r objects from a set of n objects.
  • Example of combinations: Number of ways to choose a committee from a group of people.
  • Application of combinations in real-life scenarios: Determining the number of possible combinations in a lottery game.

Probability - Permutations

  • Permutations: A permutation is an arrangement of objects in a certain order. The number of permutations of r objects from a set of n objects is given by the formula: P(n, r) = n! / (n-r)!, where n ≥ r.
  • Permutation formula: The permutation formula is used to calculate the number of ways to arrange objects in a certain order.
  • Example of permutations: Arranging a set of books on a shelf.
  • Application of permutations in real-life scenarios: Determining the number of possible arrangements in a seating plan.

Probability - Binomial Distribution

  • Binomial distribution: The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials.
  • Bernoulli trials: Bernoulli trials are random experiments with two possible outcomes - success or failure.
  • Binomial probability formula: The probability of getting exactly k successes in N trials is given by the formula: P(X=k) = (N choose k) * p^k * (1-p)^(N-k), where p is the probability of success.
  • Example of binomial distribution: Flipping a coin multiple times and counting the number of heads.
  • Properties of binomial distribution: The mean, variance, and standard deviation of a binomial distribution can be calculated using the formula: Mean = N * p, Variance = N * p * (1-p), Standard deviation = sqrt(N * p * (1-p)).

Probability - Normal Distribution

  • Normal distribution: The normal distribution is a continuous probability distribution that is symmetric about its mean and characterized by its mean and standard deviation.
  • Characteristics of normal distribution: The normal distribution is bell-shaped and symmetric, with its mean, median, and mode all coinciding at the center. It follows the 68-95-99.7 rule, where approximately 68%, 95%, and 99.7% of data falls within one, two, and three standard deviations from the mean, respectively.
  • Standard normal distribution: The standard normal distribution is a normal distribution with a mean of zero and a standard deviation of one.
  • Z-score and its calculation: The Z-score represents the number of standard deviations an individual data point is from the mean. It is calculated using the formula: Z = (X - μ) / σ, where X is the data point, μ is the mean, and σ is the standard deviation.
  • Examples of normal distribution: Heights, weights, IQ scores, etc., often follow a normal distribution pattern.