Probability

Introduction

• Axiomatic Approach: An axiom is an assumption that is a universal truth. For example, a pen and cap example.
• Sample Space: A complete explanation of an event.
  • Example: Sample space in a die throw: {1,2,3,4,5,6}
  • Sample Point: Each number on a die is called a sample point.

Rules of Probability

1. The probability of any event cannot be less than 0 or more than 1.
   • Example: Probability of an even number, for {2,4,6} on a die is 3/6 = 1/2
2. The sum of probabilities of all possible events is 1.
   • Example: The sum of probabilities of even and odd numbers is equal to 1.
3. The sum of the probability of an event is equal to the probabilities of all its sample points.

Types of Probability

• Equally Likely Events: Events that have equal probability.
  • Example: The probability of each face of a die is (1/6).
• Union of Events (A or B): (P(A \cup B) = P(A) + P(B) - P(A \cap B))
  • When there is no common: (P(A \cup B) = P(A) + P(B))
• Complement of Event (not A): (P(A') = 1 - P(A))

De Morgan's Law

• ((E \cap F)' = E' \cup F')
• ((E \cup F)' = E' \cap F')

Cards

• Deck: 52 cards consisting of 26 red and 26 black cards.
  • Red: 13 hearts and 13 diamonds
  • Black: 13 clubs and 13 spades
• Face Cards: King, Queen, Jack

Examples

1. Probability of a Diamond Card: (13/52 = 1/4)
2. Not Ace: (48/52 = 12/13)
3. Black Card: (26/52 = 1/2)
4. Not Diamond: (39/52 = 3/4)
5. Not Black Card: (26/52 = 1/2)

Example of Two Students

• Probability Anil passes the exam: (0.05)
• Probability Ashima passes the exam: (0.10)
• Probability both pass: (0.02)
• Conversation:
  1. Neither passes: (0.87)
  2. At least one does not pass: (0.98)
  3. Only one passes: (0.11)

Example of Committee

• 2 out of 4 persons are selected.
• Number of possible outcomes: (6)
• Probability:
  1. No man: (1/6)
  2. One man: (2/3)
  3. Both men: (1/6)

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These notes provide an explanation of examples and probability based on Exercise 14.2.