Probability - Origin of theory of Probability

  • Probability is a branch of mathematics that deals with the likelihood of events occurring.

  • The theory of probability has its origins in various civilizations such as ancient China, Egypt, and Greece.

  • In the 16th century, Italian mathematicians such as Gerolamo Cardano and Girolamo Tiraboschi contributed to the development of probability theory.

  • Blaise Pascal and Pierre de Fermat made significant contributions to probability theory in the 17th century.

  • The theory of probability gained further recognition and prominence through the works of mathematicians like Jakob Bernoulli and Pierre-Simon Laplace. Probability - Terminologies

  • Experiment: An activity or process that leads to the observation of certain outcomes.

  • Outcome: The result of a single performance of the experiment.

  • Sample Space: The set of all possible outcomes of an experiment, denoted by S.

  • Event: A subset of the sample space, denoted by A.

  • Probability: A numerical measure of the likelihood of an event occurring, denoted by P(A).

  • Complementary Event: The event that an event A does not occur, denoted by A’. Probability - Basic Principles

  • The probability of an event A is denoted by P(A) and lies between 0 and 1.

  • The sum of the probabilities of all possible outcomes in the sample space is equal to 1.

  • The probability of an event not occurring (complementary event) is 1 minus the probability of the event occurring.

  • The probability of the union of two events A and B is the sum of their individual probabilities minus the probability of their intersection.

  • The probability of the intersection of two independent events A and B is the product of their individual probabilities. Probability - Types of Probability

  1. Classical Probability:
    • Based on the assumption of equally likely outcomes.
    • P(A) = Number of favorable outcomes / Total number of outcomes.
    • Example: Tossing a fair coin.
  1. Empirical Probability:
    • Based on observed or collected data.
    • P(A) = Number of times event A occurred / Total number of observations.
    • Example: Rolling a die and recording the outcomes.
  1. Subjective Probability:
    • Based on personal judgment or opinions.
    • Often used in situations where objective data is unavailable.
    • Example: Predicting the outcome of a sports event. Probability - Addition Rule

  • The addition rule of probability states that the probability of the union of two non-mutually exclusive events A and B is given by:

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

  • If A and B are mutually exclusive, then P(A and B) = 0, and the formula simplifies to:

    • P(A or B) = P(A) + P(B) Probability - Conditional Probability

  • The conditional probability of an event A, given that event B has already occurred, is denoted by P(A|B).

  • The formula for conditional probability is:

    • P(A|B) = P(A and B) / P(B)

  • It helps us calculate the probability of an event A occurring, considering that event B has already occurred.

  • The conditional probability is closely related to the concept of independence between events. Probability - Multiplication Rule

  • The multiplication rule of probability allows us to calculate the probability of the intersection of two events A and B.

  • The formula for the multiplication rule is:

    • P(A and B) = P(A) * P(B|A)

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

  • The multiplication rule is used in situations where events are dependent on each other. Note: Always remember that probability is a mathematical tool used to quantify uncertainty. It helps in decision-making and understanding the likelihood of events occurring.

Slide 11:

Probability - Origin of theory of Probability

  • Probability is a branch of mathematics that deals with the likelihood of events occurring.
  • The theory of probability has its origins in various civilizations such as ancient China, Egypt, and Greece.
  • In the 16th century, Italian mathematicians such as Gerolamo Cardano and Girolamo Tiraboschi contributed to the development of probability theory.
  • Blaise Pascal and Pierre de Fermat made significant contributions to probability theory in the 17th century.
  • The theory of probability gained further recognition and prominence through the works of mathematicians like Jakob Bernoulli and Pierre-Simon Laplace.

Slide 12:

Probability - Terminologies

  • Experiment: An activity or process that leads to the observation of certain outcomes.
  • Outcome: The result of a single performance of the experiment.
  • Sample Space: The set of all possible outcomes of an experiment, denoted by S.
  • Event: A subset of the sample space, denoted by A.
  • Probability: A numerical measure of the likelihood of an event occurring, denoted by P(A).

Slide 13:

Probability - Terminologies (contd.)

  • Complementary Event: The event that an event A does not occur, denoted by A'.
  • Equally Likely Outcomes: When all outcomes in the sample space have an equal chance of occurring.
  • Favorable Outcomes: Outcomes that satisfy the conditions of a specific event.
  • Unfavorable Outcomes: Outcomes that do not satisfy the conditions of a specific event.

Slide 14:

Probability - Basic Principles

  • The probability of an event A is denoted by P(A) and lies between 0 and 1.
  • The sum of the probabilities of all possible outcomes in the sample space is equal to 1.
  • The probability of an event not occurring (complementary event) is 1 minus the probability of the event occurring.
  • The probability of the union of two events A and B is the sum of their individual probabilities minus the probability of their intersection.

Slide 15:

Probability - Basic Principles (contd.)

  • The probability of the intersection of two independent events A and B is the product of their individual probabilities.
  • The Law of Large Numbers states that as the number of experiments increases, the experimental probability of an event approaches the theoretical probability.
  • Experimental Probability is based on observed data, while Theoretical Probability is based on mathematical calculations.

Slide 16:

Probability - Types of Probability

  1. Classical Probability:
    • Based on the assumption of equally likely outcomes.
    • P(A) = Number of favorable outcomes / Total number of outcomes.
    • Example: Tossing a fair coin.
  1. Empirical Probability:
    • Based on observed or collected data.
    • P(A) = Number of times event A occurred / Total number of observations.
    • Example: Rolling a die and recording the outcomes.

Slide 17:

Probability - Types of Probability (contd.)

  1. Subjective Probability:
    • Based on personal judgment or opinions.
    • Often used in situations where objective data is unavailable.
    • Example: Predicting the outcome of a sports event.
  1. Conditional Probability:
    • The probability of an event A occurring, given that event B has already occurred.
    • Denoted by P(A|B), where P(B) ≠ 0.
    • Example: Drawing cards from a deck without replacement.

Slide 18:

Probability - Addition Rule

  • The addition rule of probability states that the probability of the union of two non-mutually exclusive events A and B is given by:
    • P(A or B) = P(A) + P(B) - P(A and B)
  • If A and B are mutually exclusive, then P(A and B) = 0, and the formula simplifies to:
    • P(A or B) = P(A) + P(B)

Slide 19:

Probability - Addition Rule (contd.)

  • The addition rule can be extended to more than two events:
    • P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C)
  • The rule can be used to find the probability of events occurring together or separately.

Slide 20:

Probability - Multiplication Rule

  • The multiplication rule of probability allows us to calculate the probability of the intersection of two events A and B.
  • The formula for the multiplication rule is:
    • P(A and B) = P(A) * P(B|A)
  • Here, P(B|A) represents the conditional probability of event B occurring, given that event A has already occurred.
  • The multiplication rule is used in situations where events are dependent on each other.

Slide 21:

Probability - Multiplication Rule (contd.)

  • The multiplication rule can be extended to more than two events:
    • P(A and B and C) = P(A) * P(B|A) * P(C|A and B)
  • This rule can be used to find the probability of multiple events occurring together.
  • Example:
    • A bag contains 3 red balls and 2 blue balls. If 2 balls are drawn without replacement, find the probability of drawing a red ball followed by a blue ball.
    • Solution:
      • P(Red then Blue) = P(Red) * P(Blue|Red)
      • P(Red then Blue) = (3/5) * (2/4)
      • P(Red then Blue) = 6/20
      • P(Red then Blue) = 3/10

Slide 22:

Probability - Bayes’ Theorem

  • Bayes’ Theorem is a formula used to calculate conditional probabilities.
  • It is named after Thomas Bayes, an 18th-century British mathematician.
  • The formula for Bayes’ Theorem is:
    • P(A|B) = (P(B|A) * P(A)) / P(B)
  • Here, P(A|B) is the probability of event A given that event B has already occurred.
  • P(B|A) is the conditional probability of event B occurring, given that event A has already occurred.
  • P(A) and P(B) are the probabilities of events A and B occurring, respectively.

Slide 23:

Probability - Bayes’ Theorem (contd.)

  • Bayes’ Theorem can be used to update our prior beliefs (prior probabilities) based on new evidence.
  • It is often applied in the field of statistics, data analysis, and machine learning.
  • The theorem can be used to make predictions or inference about events or hypotheses.
  • Example:
    • A blood test for a certain disease is known to correctly identify 98% of the patients who have the disease and 96% of the patients who do not have the disease. If 1% of the population has the disease, what is the probability that a person has the disease given that the test result is positive?
    • Solution:
      • Let A be the event that a person has the disease.
      • Let B be the event that the test result is positive.
      • P(A) = 0.01 (given)
      • P(B|A) = 0.98 (disease correctly identified)
      • P(B|A’) = 0.04 (healthy person wrongly identified)
      • P(A|B) = (P(B|A) * P(A)) / (P(B|A) * P(A) + P(B|A’) * P(A’))
      • P(A|B) = (0.98 * 0.01) / (0.98 * 0.01 + 0.04 * 0.99)
      • P(A|B) ≈ 0.20

Slide 24:

Probability - Permutations

  • Permutations refer to the different ways in which a set of objects can be arranged.
  • The number of permutations of n objects taken r at a time can be calculated using the formula:
    • P(n, r) = n! / (n - r)!
  • Here, n is the total number of objects and r is the number of objects taken at a time.
  • The exclamation mark (!) denotes factorial, which is the product of all positive integers less than or equal to a given number.
  • Example:
    • How many different ways can 3 students be arranged in a line?
    • Solution:
      • P(3, 3) = 3! / (3 - 3)! = 3! / 0! = 3 * 2 * 1 = 6

Slide 25:

Probability - Combinations

  • Combinations refer to the different ways in which a subset of objects can be selected from a larger set, without considering the order.
  • The number of combinations of n objects taken r at a time can be calculated using the formula:
    • C(n, r) = n! / (r! * (n - r)!)
  • Here, n is the total number of objects and r is the number of objects taken at a time.
  • The exclamation mark (!) denotes factorial, which is the product of all positive integers less than or equal to a given number.
  • Example:
    • In a group of 7 people, how many different committees of 3 people can be formed?
    • Solution:
      • C(7, 3) = 7! / (3! * (7 - 3)!) = 7! / (3! * 4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35

Slide 26:

Probability - Expected Value

  • Expected value (also known as the mean or average) is a measure of central tendency.
  • It represents the average outcome of a random experiment after a large number of trials.
  • The expected value of a discrete random variable X can be calculated using the formula:
    • E(X) = Σ(x * P(x))
  • Here, x is the value of the random variable, P(x) is the probability of X taking the value x, and Σ represents the sum over all possible values of X.
  • Example:
    • Consider the following probability distribution for a random variable X: x | 1 | 2 | 3 P | 0.4 | 0.3 | 0.3
    • Find the expected value of X.
    • Solution:
      • E(X) = (1 * 0.4) + (2 * 0.3) + (3 * 0.3) = 0.4 + 0.6 + 0.9 = 1.9

Slide 27:

Probability - Variance and Standard Deviation

  • Variance and standard deviation are measures of dispersion in a probability distribution.
  • Variance (σ^2) is calculated as the average of the squared deviations from the expected value.
  • Standard deviation (σ) is the square root of the variance.
  • The formulas for variance and standard deviation are:
    • Var(X) = Σ((x - E(X))^2 * P(x))
    • SD(X) = √Var(X)
  • Example:
    • Consider the following probability distribution for a random variable X: x | 1 | 2 | 3 P | 0.4 | 0.3 | 0.3
    • Find the variance and standard deviation of X.
    • Solution:
      • E(X) = 1.9 (from the previous example)
      • Var(X) = ((1 - 1.9)^2 * 0.4) + ((2 - 1.9)^2 * 0.3) + ((3 - 1.9)^2 * 0.3) = 0.19
      • SD(X) = √Var(X) ≈ √0.19 ≈ 0.44

Slide 28:

Probability - Probability Distributions

  • A probability distribution is a function that assigns probabilities to the possible outcomes of a random experiment.
  • It provides a mathematical representation of the likelihood of each outcome.
  • Common probability distributions include:
    • Discrete Probability Distribution (e.g., Binomial, Poisson)
    • Continuous Probability Distribution (e.g., Normal, Exponential)
  • Each probability distribution has its own set of properties, characteristics, and formulas for calculating probabilities.

Slide 29:

Probability - Binomial Distribution

  • The binomial distribution is a probability distribution for a discrete random variable, where each trial has two possible outcomes: success or failure.
  • It is characterized by two parameters:
    • n: the number of trials
    • p: the probability of success in a single trial
  • The probability mass function (PMF) of the binomial distribution is given by:
    • P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
    • Here, X is the random variable, k is the number of successes, C(n, k) is the number of combinations of n objects taken k at a time, and p is the probability of success.
  • Example:
    • A biased coin has a 0.7 probability of landing on heads. If the coin is flipped 4 times, find the probability of getting exactly 3 heads.
    • Solution:
      • P(X = 3) = C(4, 3) * 0.7^3 * (1 - 0.7)^(4 - 3) = 4 * 0.7^3 * 0.3^1 = 0.4116

Slide 30:

Probability - Normal Distribution

  • The normal distribution (also known as the Gaussian distribution) is a continuous probability distribution.
  • It is characterized by its mean (μ) and standard deviation (σ).
  • The shape of the normal distribution is symmetric, bell-shaped, and approximately follows the 68-95-99.7 rule (Empirical Rule).
  • The probability density function (PDF) of the normal distribution is given by:
    • f(x) = (1 / (σ√(2π))) * e^(-((x - μ)^2) / (2σ^2))
    • Here, x is the random variable, μ is the mean, σ is the standard deviation, π is a mathematical constant (approximately 3.14159), and e is the base of the natural logarithm (approximately 2.71828).
  • Example:
    • Consider a normal distribution with a mean of 50 and