UNIT-2.pdf

ABE 23 – Engineering Data Analysis UNIT II PROBABILITY Audry Llaban Anacio Department of Agricultural and Biosystems Engineering College of Engineering Central Mindanao University Probability - is a branch of mathematics that measures...

ABE 23 – Engineering Data Analysis UNIT II PROBABILITY Audry Llaban Anacio Department of Agricultural and Biosystems Engineering College of Engineering Central Mindanao University Probability - is a branch of mathematics that measures the likelihood or chance of an event occurring. Key Concepts of Probability: Probability Definition: Probability is defined as a number between 0 and 1, where: 0 represents an impossible event (the event will never happen) 1 represents a certain event (the event will always happen). A probability closer to 1 indicates a higher likelihood of the event occurring, while a probability closer to 0 indicates a lower likelihood. The probability of an event A is usually written as P(A).  Sample Space and Relationships among Events A. Sample Space - The sample space (S) is the set of all possible outcomes of a random experiment. Example: Measuring the daily rainfall (in mm) in a specific region. Sample Space: S = {0, 1, 2, 3,..., 100} mm B. Events - An event (E) is any subset of the sample space. Example: Event A: Rainfall is less than 10 mm. Event B: Rainfall is between 20 mm and 30 mm. Sample Space and Relationships among Events C. Relationships among Events 1. Complementary events – The complement of an event A (denoted by A’) includes all outcomes in the sample space that are not in A. Example: If A is the event "rainfall is less than 10 mm," then A' is the event "rainfall is 10 mm or more." 2. Union of events - The union of two events A and B (denoted by A ∪ B) includes all outcomes that are in A, in B, or in both. Example: If A is "rainfall is less than 10 mm" and B is "rainfall is more than 50 mm,“ then A ∪ B is the event "rainfall is less than 10 mm or more than 50 mm." Sample Space and Relationships among Events C. Relationships among Events 3. Intersection of events - The intersection of two events A and B (denoted by A ∩ B) includes all outcomes that are in both A and B. Example: If A is "rainfall is more than 20 mm" and B is "rainfall is less than 30 mm," then A ∩ B is the event "rainfall is between 20 mm and 30 mm.“ 4. Mutually exclusive events - Two events A and B are mutually exclusive if they cannot both occur at the same time (A ∩ B = ∅). Example: If A is "rainfall is less than 10 mm" and B is "rainfall is more than 50 mm," then A and B are mutually exclusive.  Counting Rules useful in Probability A. Fundamental Counting Principle If an event can occur in m ways and a second event can occur independently of the first in n ways, then the two events together can occur in m × n ways. Example: Scenario: Planting two types of crops (corn and wheat) in two different fields. Number of ways to plant corn: 3 Number of ways to plant wheat: 2 Total number of ways to plant both crops: 3 × 2 = 6  Counting Rules useful in Probability B. Permutations A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is given by: 𝑛! 𝑃 𝑛, 𝑟 = 𝑛−𝑟 ! Example: Scenario: Arranging 3 different types of fertilizers in a trial with 5 plots. 5! 120 Number of permutations: 𝑃 5,3 = = = 𝟔𝟎 5−3 ! 2  Counting Rules useful in Probability C. Combinations A combination is a selection of objects without regard to order. The number of combinations of n objects taken r at a time is given by: 𝑛! C 𝑛, 𝑟 = 𝑟! 𝑛−𝑟 ! Example: Scenario: Selecting 3 crop varieties out of 5 for an experiment.. 5! 120 Number of permutations: C 5,3 = = = 𝟏𝟎 3! 5−3 ! 6×2  Rules of Probability A. Basic Probability Rules 1. Probability of an Event - The probability of an event A, denoted by P(A), is the measure of the likelihood that A will occur. 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 Formula: 𝑃 𝐴 = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 Example: Probability of selecting a defective sensor from a batch of 100 sensors with 5 defective sensors: 5 𝑃 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 = = 𝟎. 𝟎𝟓 100  Rules of Probability A. Basic Probability Rules 2. Complement Rule- The probability of the complement of an event 𝐴 𝐴′ is given by: Formula: 𝑃 𝐴′ = 1 − 𝑃(𝐴) Example: If P(A) = 0.3, then 𝑃 𝐴′ = 1 − 0.3 = 𝟎. 𝟕  Rules of Probability A. Basic Probability Rules 2. Addition Rule - For any two events A and B: 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵) - If A and B are mutually exclusive, 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 Example: Probability of rainfall being less than 10mm P(A) = 0.4, or more than 50mm P(B) = 0.2, since these events are mutually exclusive: 𝑃 𝐴 ∪ 𝐵 = 0.4 + 0.2 = 𝟎. 𝟔  Rules of Probability A. Basic Probability Rules 3. Multiplication Rule - For any two events A and B: 𝑃 𝐴 ∩ 𝐵 = 𝑃(𝐴) × 𝑃(𝐵ȁ𝐴) - If A and B are independent, 𝑃 𝐴 ∩ 𝐵 = 𝑃(𝐴) × 𝑃(𝐵) Example: Probability of rainfall being less than 10mm P(A) = 0.4, and temperature being below 20°C P(B) = 0.5 if these events are independent: 𝑃 𝐴 ∩ 𝐵 = 0.4 × 0.5 = 𝟎. 𝟐