Probability - Making new events using set theory

Slide 1: Introduction to Making New Events using Set Theory

  • In probability theory, we often need to create new events using set theory
  • Set theory allows us to combine or manipulate events to obtain different outcomes
  • In this lecture, we will learn how to make new events using set theory in probability

Slide 2: Basic Concepts in Set Theory

  • In set theory, we deal with sets, which are collections of objects
  • Objects in a set are called elements
  • We represent a set by listing its elements inside curly braces, separated by commas
    • Example: A = {1, 2, 3, 4, 5} represents a set named A with elements 1, 2, 3, 4, and 5

Slide 3: Union of Two Sets

  • The union of two sets A and B, denoted as A ∪ B, is the set that contains all the elements of A and B
  • A ∪ B = {x : x belongs to A or x belongs to B}
  • Example: A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}

Slide 4: Intersection of Two Sets

  • The intersection of two sets A and B, denoted as A ∩ B, is the set that contains all the common elements of A and B
  • A ∩ B = {x : x belongs to A and x belongs to B}
  • Example: A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}

Slide 5: Complement of a Set

  • The complement of a set A, denoted as A’, is the set that contains all the elements that are not in A but are in the universal set
  • A’ = {x : x is in the universal set and x is not in A}
  • Example: A = {1, 2, 3, 4, 5} and the universal set is {1, 2, 3, 4, 5, 6, 7}, then A’ = {6, 7}

Slide 6: Difference between Two Sets

  • The difference between two sets A and B, denoted as A - B, is the set that contains all the elements of A that are not in B
  • A - B = {x : x belongs to A and x does not belong to B}
  • Example: A = {1, 2, 3} and B = {3, 4, 5}, then A - B = {1, 2}

Slide 7: Disjoint Sets

  • Two sets A and B are said to be disjoint if their intersection is an empty set
  • A ∩ B = {}
  • Example: A = {1, 2, 3} and B = {4, 5, 6} are disjoint sets

Slide 8: Example Problem 1

  • Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}
  • Find A ∪ B, A ∩ B, A’, and B'
  • Solution:
    • A ∪ B = {1, 2, 3, 4, 5, 6}
    • A ∩ B = {3, 4}
    • A’ = {}
    • B’ = {}

Slide 9: Example Problem 2

  • Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}
  • Find A - B and B - A
  • Solution:
    • A - B = {1, 2}
    • B - A = {5, 6}

Slide 10: Summary and Conclusion

  • Set theory allows us to create new events in probability
  • We can combine sets using union and intersection operations
  • Complement and difference operations help us manipulate sets further
  • Understanding set theory is essential in probability for creating new events

Slide 11: Union and Intersection of Multiple Sets

  • We can extend the concepts of union and intersection to more than two sets
  • Union of Multiple Sets:
    • The union of three sets A, B, and C, denoted as A ∪ B ∪ C, is the set that contains all the elements of A, B, and C
    • A ∪ B ∪ C = {x : x belongs to A or x belongs to B or x belongs to C}
  • Intersection of Multiple Sets:
    • The intersection of three sets A, B, and C, denoted as A ∩ B ∩ C, is the set that contains all the elements that are common to A, B, and C
    • A ∩ B ∩ C = {x : x belongs to A and x belongs to B and x belongs to C}
  • Example: A = {1, 2, 3}, B = {2, 3, 4}, and C = {3, 4, 5}
    • A ∪ B ∪ C = {1, 2, 3, 4, 5}
    • A ∩ B ∩ C = {3}

Slide 12: Complement of Multiple Sets

  • Similar to complement of a single set, we can find the complement of multiple sets
  • The complement of multiple sets A, B, and C, denoted as (A ∪ B ∪ C)’, is the set that contains all the elements that are not in A, B, or C but are in the universal set
  • (A ∪ B ∪ C)’ = {x : x is in the universal set and x is not in A, B, or C}
  • Example: A = {1, 2, 3}, B = {2, 3, 4}, C = {3, 4, 5}, and the universal set is {1, 2, 3, 4, 5, 6, 7}
    • (A ∪ B ∪ C)’ = {6, 7}

Slide 13: Venn Diagrams

  • Venn diagrams are graphical representations of sets
  • They help us visualize how sets are related and how they overlap
  • In a Venn diagram, each set is represented by a circle or an oval, and the elements of the set are placed inside the circle
  • The overlapping region between circles represents the common elements between sets
  • Venn diagrams are useful in understanding set operations and solving problems involving multiple sets

Slide 14: Example Problem 3

  • Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {4, 5, 6, 7}
  • Represent these sets using a Venn diagram
  • Solution:
    • Draw three circles to represent sets A, B, and C
    • Place the elements of each set inside the corresponding circle
    • The overlapping regions represent the common elements between sets

Slide 15: Sample Venn Diagram

  • A Venn diagram representing sets A, B, and C
  • Venn Diagram
  • In this Venn diagram, the overlapping region between circles A, B, and C represents the common element 4

Slide 16: Disjoint Sets and Intersection

  • If the intersection of two sets A and B is an empty set, then the sets are called disjoint sets
  • Disjoint sets have no elements in common
  • Example: A = {1, 2, 3}, B = {4, 5, 6}
    • A ∩ B = {} (empty set)
    • A and B are disjoint sets

Slide 17: Subset and Superset

  • If every element of set A is also an element of set B, then we say that A is a subset of B, denoted as A ⊆ B
  • Example: A = {1, 2}, B = {1, 2, 3}
    • A ⊆ B
  • If set A is a subset of set B and B is not a subset of A, then we say that A is a proper subset of B, denoted as A ⊂ B
  • Example: A = {1, 2}, B = {1, 2, 3}
    • A ⊂ B

Slide 18: Probability of Events using Set Theory

  • Set theory plays a crucial role in calculating probabilities of events
  • Probability is essentially measuring the likelihood of an event occurring
  • We can use set operations to calculate probabilities by counting the number of favorable outcomes and total possible outcomes

Slide 19: Example Problem 4

  • Let S be the sample space of rolling a fair die
  • Define two events A and B as follows:
    • A: An even number appears on the die
    • B: A prime number appears on the die
  • Find P(A), P(B), and P(A ∪ B)
  • Solution:
    • A = {2, 4, 6} (3 even numbers out of 6 possible outcomes)
    • B = {2, 3, 5} (3 prime numbers out of 6 possible outcomes)
    • P(A) = 3/6 = 1/2
    • P(B) = 3/6 = 1/2
    • P(A ∪ B) = 4/6 = 2/3

Slide 20: Summary and Conclusion

  • Set theory provides useful tools for creating new events and manipulating sets in probability
  • Union, intersection, complement, and difference operations are often used to combine or analyze events
  • Venn diagrams help visualize the relationships between sets
  • Understanding set theory is essential in probability for calculating probabilities of events
  1. Slide: Conditional Probability
  • Conditional probability is the probability of an event A occurring given that another event B has already occurred
  • It is denoted as P(A|B), read as “probability of A given B”
  • The formula for conditional probability is:
    • P(A|B) = P(A and B) / P(B)
  • Example: Suppose we roll two fair dice, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6}. Find P(A|B).
    • P(A and B) = {2, 4, 6} = 3/36 = 1/12
    • P(B) = {2, 4, 6} = 3/36 = 1/12
    • P(A|B) = (1/12) / (1/12) = 1
  1. Slide: Independent Events
  • Two events A and B are said to be independent if the occurrence of one event does not affect the probability of the other event
  • Mathematically, if P(A|B) = P(A), then A and B are independent
  • Example: Tossing a fair coin and rolling a fair die are independent events since the outcome of one does not affect the other
  1. Slide: Mutually Exclusive Events
  • Two events A and B are said to be mutually exclusive if they cannot occur at the same time
  • Mathematically, if A and B are mutually exclusive, then P(A and B) = 0
  • Example: Tossing a coin and getting a head (A) and getting a tail (B) are mutually exclusive events
  1. Slide: Addition Rule of Probability
  • The addition rule of probability states that the probability of the union of two events A and B is equal to the sum of their individual probabilities minus the probability of their intersection
  • Mathematically, P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  • Example: A fair die is rolled. Find the probability of getting an even number (A) or a prime number (B).
    • P(A) = 3/6 = 1/2 (3 even numbers out of 6 possible outcomes)
    • P(B) = 3/6 = 1/2 (3 prime numbers out of 6 possible outcomes)
    • P(A ∩ B) = 0 (no number is both even and prime)
    • P(A ∪ B) = 1/2 + 1/2 - 0 = 1
  1. Slide: Multiplication Rule of Probability
  • The multiplication rule of probability states that the probability of the intersection of two independent events A and B is equal to the product of their individual probabilities
  • Mathematically, P(A ∩ B) = P(A) * P(B)
  • Example: A fair coin is tossed twice. Find the probability of getting two heads (A) and two tails (B).
    • P(A) = 1/2 * 1/2 = 1/4 (probability of getting a head on the first and second toss)
    • P(B) = 1/2 * 1/2 = 1/4 (probability of getting a tail on the first and second toss)
    • P(A ∩ B) = 0 (no outcome satisfies both A and B)
  1. Slide: Complementary Events
  • Complementary events are two events that together cover all possible outcomes
  • The probability of an event A and its complement A’ is always equal to 1
  • Mathematically, P(A) + P(A’) = 1
  • Example: The probability of getting a head (A) on a fair coin is 1/2, and the probability of getting a tail (A’) is also 1/2.
  1. Slide: Sample Space and Event Space
  • The sample space of an experiment is the set of all possible outcomes of the experiment
  • It is denoted as S
  • Example: For rolling a single fair die, the sample space is S = {1, 2, 3, 4, 5, 6}
  • An event space is a subset of the sample space that contains specific outcomes of interest
  • Example: In the same die experiment, the event space for getting an odd number is A = {1, 3, 5}
  1. Slide: Law of Total Probability
  • The law of total probability states that if an experiment results in a partition of the sample space into mutually exclusive events A₁, A₂, …, An, then the probability of an event B is given by the sum of the probabilities of B given each Ai, weighted by the probability of Ai
  • Mathematically, P(B) = P(B|A₁) * P(A₁) + P(B|A₂) * P(A₂) + … + P(B|An) * P(An)
  • Example: Two factories produce electronic components A and B, with probabilities A = 0.2 and B = 0.8. The defect rate for components manufactured by A is 0.1, while the defect rate for components manufactured by B is 0.2. Find the probability that a randomly selected component is defective.
    • P(defective|A) = 0.1
    • P(defective|B) = 0.2
    • P(defective) = 0.1 * 0.2 + 0.2 * 0.8 = 0.18
  1. Slide: Bayes’ Theorem
  • Bayes’ theorem is a formula that allows us to update the probability of an event based on new information or evidence
  • Mathematically, it is given by:
    • P(A|B) = (P(B|A) * P(A)) / P(B)
  • Example: A blood test for a certain disease is 98% accurate, with a false positive rate of 3%. If the prevalence of the disease in the population is 0.5%, find the probability that a person has the disease given that the test is positive.
    • P(A) = 0.005 (prevalence of the disease)
    • P(B|A) = 0.98 (accuracy of the test)
    • P(B|A’) = 0.03 (false positive rate)
    • P(A|B) = (0.98 * 0.005) / (0.98 * 0.005 + 0.03 * 0.995) ≈ 0.14
  1. Slide: Summary and Conclusion
  • Set theory and probability theory go hand in hand, allowing us to analyze and calculate probabilities of events
  • Conditional probability, independent events, mutually exclusive events, and probability rules play a vital role in probability calculations
  • Understanding these concepts and applying them correctly is essential in problem-solving for probability-related questions
  • Keep practicing and applying these concepts to gain a better understanding and mastery of probability