probability, binomial and poisson Distribution

Lecture 6: Probability

Basic Concepts

• Probability Definition: A measure of the likelihood of the occurrence of an event, typically expressed as a number between 0 (impossible) and 1 (certain).

• Trial and Event: In probability, a trial is an experiment that produces an outcome which can vary. An example is seed germination, where seeds can either germinate or not.

• Outcomes: Each result of a trial is known as an event.

Sample Space (S)

• Definition: The set of all possible outcomes of an experiment.

  • Example with seeds: Possible outcomes are {0, 1, 2, 3, 4, 5} seeds germinating.

Types of Events

• Exhaustive Events: Total possible outcomes in a given trial.

  • Example with a die: The six faces (1, 2, 3, 4, 5, 6) represent exhaustive cases.

• Favourable Events: Outcomes that directly relate to the event in question.

• Mutually Exclusive Events: Events that cannot occur simultaneously. Example: Seed germination (either germinates or does not).

• Equally Likely Events: Events that have the same chance of occurring.

• Independent Events: Events where the occurrence of one does not affect the other. Example: The germination of one seed does not affect another.

• Dependent Events: The outcome of one event affects the occurrence of another. Example: Card drawing without replacement.

Definitions of Probability

• Probability (Mathematical/ Classical): For an event A, if there are n exhaustive cases with m favourable to A, then the probability is given by P(A) = m/n.

  • If m = 0, then P(A) = 0 (impossible event).

  • If m = n, then P(A) = 1 (certain event).

• Statistical (Empirical) Probability: If an event A occurs m times in n trials, then the probability is P(A) = m/n.

Axioms of Probability

1. P(E) ranges from 0 to 1; if an event cannot take place, its probability is 0; if it is certain, its probability is 1.

2. P(S) (entire sample space) = 1.

3. If A and B are mutually exclusive, then P(A or B) = P(A) + P(B).

Examples of Probability

• Dice Example: For two dice, total outcomes = 36; calculating P(Sum = 6) involves favorable outcomes like (1,5), (2,4), etc.

• Card Example: When drawing cards from a pack of 52 cards:

  • Getting a King: 4 kings out of 52, hence P(King) = 4/52.

Conditional Probability

• Definition: P(A given B), represents the probability of event A occurring given that B has occurred.

Theorems of Probability

1. Addition Theorem: For events A and B,

   • If not mutually exclusive: P(A or B) = P(A) + P(B) - P(A and B).

   • If mutually exclusive: P(A or B) = P(A) + P(B).

2. Multiplication Theorem:

   • For dependent events: P(A and B) = P(A) * P(B|A).

   • For independent events: P(A and B) = P(A) * P(B).

Lecture 7: Theoretical Distributions

Types of Distributions

1. Binomial Distribution

   • Directly related to Bernoulli trials where n consists of independent trials and the probability of success is constant.

   • Probability Function: For x successes in n trials is given by P(X = x) = (nCx)(p^x)(q^(n-x)) where q = 1 - p.

   • Properties:

     • Mean = np

     • Variance = npq

   • Applications: Used in quality control and medical success rate studies.

Examples

• Tossing coins: To calculate probabilities based on outcomes when tossing coins or drawing from a deck.

Poisson Distribution

• Description: A discrete probability distribution, primarily used for rare events per time unit.

  • Probability Function: P(x) = (e^(-λ) * λ^x) / x! where λ is the average rate of occurrence.

  • Conditions: Derived from limiting conditions of binomial distribution when n is large and p is small.

  • Properties: Mean = Variance = λ.

  • Applications: Applicable in various fields for counting rare events.

Normal Distribution

• Characterization: A continuous distribution characterized by its bell shape; important in statistics due to the central limit theorem.

• Probability Function: Defined by parameters mean (µ) and variance (σ^2).

• Properties:

  • Symmetric around the mean.

  • The area under the curve totals 1, and mean = median = mode.

Questions and Answers Summary

• Probability is expressed as a ratio, percentage, or proportion (answer: all the above).

• A sure event has a probability of 1 (True).

• The total area under the normal distribution curve is one (True).