ABE 23 - Engineering Data Analysis Unit II

Probability Overview

• Probability measures the likelihood of an event occurring, expressed as a value between 0 and 1.
• A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event.
• Values closer to 1 denote a higher likelihood, while values closer to 0 denote a lower likelihood.
• The probability of event A is represented as P(A).

Sample Space and Events

• The sample space (S) encompasses all possible outcomes of a random experiment.
  • Example: Daily rainfall in mm could have the sample space S = {0, 1, 2, ..., 100} mm.
• An event (E) is a subset of the sample space.
  • Example: Event A could be "rainfall is less than 10 mm".

Relationships Among Events

• Complementary Events: The complement of an event A (denoted A’) consists of outcomes not in A.

  • Example: If A is "rainfall < 10 mm," then A' is "rainfall ≥ 10 mm".

• Union of Events: The union of events A and B (denoted A ∪ B) includes outcomes in A, B, or both.

  • Example: A = "rainfall < 10 mm" and B = "rainfall > 50 mm" gives A ∪ B as "rainfall < 10 mm or > 50 mm".

• Intersection of Events: The intersection of events A and B (denoted A ∩ B) includes outcomes in both A and B.

  • Example: A = "rainfall > 20 mm" and B = "rainfall < 30 mm" leads to A ∩ B as "rainfall between 20 mm and 30 mm".

• Mutually Exclusive Events: Events A and B are mutually exclusive if they cannot occur simultaneously (A ∩ B = ∅).

  • Example: If A = "rainfall < 10 mm" and B = "rainfall > 50 mm," then they cannot happen at once.

Counting Rules in Probability

• Fundamental Counting Principle: If event 1 can occur in m ways and event 2 independently in n ways, they can occur together in m × n ways.

  • Example: Planting 3 corn types and 2 wheat types allows for 3 × 2 = 6 combinations.

• Permutations: Arrangements of objects where order matters. The number of permutations of n objects taken r at a time is given by n! / (n−r)!.

  • Example: Arranging 3 fertilizers in 5 plots yields P(5,3) = 60.

• Combinations: Selections of objects where order does not matter. The number of combinations of n objects taken r at a time is given by n! / (r!(n−r)!).

  • Example: Selecting 3 varieties from 5 for an experiment results in C(5,3) = 10.

Basic Probability Rules

• Probability of an Event: Probability of event A is calculated as the number of favorable outcomes divided by the total number of possible outcomes.
  • Example: For a batch of 100 sensors with 5 defective, P(defective) = 5/100 = 0.05.