Probability Distributions and Examples

Binomial Distribution

  • Definition: Models the number of successes in a fixed number of independent Bernoulli trials.
  • Characteristics:
  • Each trial has two outcomes: success or failure.
  • Probability of success remains constant.
  • Trials are independent.

Replacement in Sampling

  • With Replacement: Each selection is independent. Example: Picking a ball and putting it back before the next pick.
  • Without Replacement: Each selection affects subsequent selections, making them dependent. Example: Picking a ball and not returning it influences the next pick.

Hypergeometric Distribution

  • Use when drawing samples without replacement.
  • Example Problem:
  • Items are divided into two types (e.g., type A and type B).
  • Given total items (A + B) and the number to sample (n), find the probability of selecting a certain number of type A items.
  • Formula used is derived from combinations:
  • Probability of getting a number of type A items = (C(A, x) * C(B, n - x)) / C(A + B, n)
  • where C(a, b) = a!/(b!(a - b)!) is the combination function.

Possible Values of x

  • The variable x represents the number of successful outcomes (type A items) in n trials.
  • Values for x range from 0 up to the minimum of n and A, where A is the total number of type A items in the population.
  • Example: If A = 4, possible x values when sampling n = 3 range from 0 to 3.

Probability Examples

  • Compute probabilities using hypergeometric distribution formula. Example of selecting students where:
  • A = 7 males, B = 5 females, n = 4 selected.
Case of Selecting Students
  • Problem: Find probabilities of selecting at least three males from four selected students:
  • Calculate probability for x = 3 and x = 4 and sum them up.

Survey Example

  • If 5 dog owners are surveyed, and 3 out of 10 reported a specific behavior, model the probability of exactly 2 reporting this behavior:
  • Use the hypergeometric formula with A = 3, B = 7 (non-reported), n = 5.

Quality Control Example

  • For checking defects in a production lot:
  • If selecting items and rejecting based on defects, calculate the probability of finding more than a set number of defective items.

Geometric Distribution

  • Difference from binomial distribution:
  • Binomial: Success after n trials.
  • Geometric: Trials continue until the first success.
Geometric Probability Model
  • Example: A coin is flipped until a head appears.
  • Define p (probability of success) and calculate the probability for n trials until success.
  • Formula: P(X=k) = (1-p)^(k-1) * p for the first success on trial k.

Statistical Examples Using Geometric Distribution

  • Situational Problems (e.g., blood types in a population):
  • Example of finding the probability that the first person with a certain trait appears at a specific trial (using p and (1-p)).