1. Algebra of Events

Since events are simply subsets of a sample space, we can use the familiar operations from Set Theory (like union, intersection, and complement) to combine or modify them. This application of set operations to events is known as the algebra of events. Visualizing these operations with Venn diagrams is extremely helpful.

Set Theory and Probability: A Dictionary

To make the connection clear, here's a direct comparison between probability terms and set theory terms. Thinking about events as sets is the key to understanding the rules. 🗝️

  • Sample Space (S) is the Universal Set (U).
  • An Event (E) is a Subset (E) of the sample space.
  • The Impossible Event is the Empty Set (∅).
  • The Sure Event is the entire Sample Space (S).
  • The event 'not E' is the Complement of E (E').
  • The event 'A or B' is the Union of A and B (A ∪ B).
  • The event 'A and B' is the Intersection of A and B (A ∩ B).
  • Mutually Exclusive Events are Disjoint Sets.

Set Theory Example: Let the Universal Set be U={1,2,3,4,5,6}U = \{1, 2, 3, 4, 5, 6\}. Let subset A={2,4,6}A = \{2, 4, 6\} and subset B={5,6}B = \{5, 6\}.

  • Complement of A: A′=U−A={1,3,5}A' = U - A = \{1, 3, 5\}.
  • Union of A and B: A∪B={2,4,5,6}A \cup B = \{2, 4, 5, 6\}.
  • Intersection of A and B: A∩B={6}A \cap B = \{6\}.

This is exactly analogous to the probability examples that follow.

A. Complementary Event ('not' E)

The complement of an event E, denoted by E' or Ec^c, represents the event that 'E does not happen'. It includes all outcomes in the sample space SS that are not in the event E.

  • In Set Notation: E' = S−ES - E.
  • Venn Diagram: The complement E' is everything in the sample space rectangle that is outside the circle representing event E.
  • Die Roll Example 🎲: If E is 'getting an even number' (E = {2, 4, 6}), then the complementary event E' is 'not getting an even number', which is E' = {1, 3, 5}.
  • Card Draw Example 🃏: If A is 'drawing a King', then A' is 'not drawing a King' (i.e., drawing any of the other 48 cards).

B. The Event 'A or B' (Union)

The event 'A or B', denoted by A∪BA \cup B, represents the occurrence of at least one of the events. It includes all outcomes that are in A, in B, or in both.

  • Venn Diagram: The union A ∪ B is the total area covered by both circles A and B, including their overlapping part.
  • Die Roll Example 🎲: If A is 'getting an even number' ({2, 4, 6}) and B is 'getting a number greater than 4' ({5, 6}), then 'A or B' is A∪B={2,4,5,6}A \cup B = \{2, 4, 5, 6\}.
  • Card Draw Example 🃏: If A is 'drawing a King' and B is 'drawing a Heart', then A∪BA \cup B is the event 'drawing a King or a Heart'. This set includes the King of Hearts, the other 3 Kings, and the other 12 Hearts.

C. The Event 'A and B' (Intersection)

The event 'A and B', denoted by A∩BA \cap B, represents the simultaneous occurrence of both events. It includes only the outcomes that are common to both A and B.

  • Venn Diagram: The intersection A ∩ B is the overlapping area of the two circles representing A and B.
  • Die Roll Example 🎲: Using the same events A and B, 'A and B' is the event 'getting a number that is both even AND greater than 4'. The only such outcome is 6, so A∩B={6}A \cap B = \{6\}.
  • Card Draw Example 🃏: If A is 'drawing a King' and B is 'drawing a Heart', then A∩BA \cap B is the event 'drawing a King that is also a Heart', which is the King of Hearts.

D. The Event 'A but not B'

The event 'A but not B', denoted by A−BA - B or A∩B′A \cap B', represents the event that A occurs but B does not. It includes all outcomes that are in A but excludes any that are also in B.

  • Venn Diagram: A - B is the part of circle A that does not overlap with circle B.
  • Die Roll Example 🎲: Using the same events A and B, 'A but not B' is the event 'getting an even number that is NOT greater than 4'. This is A−B={2,4}A - B = \{2, 4\}.
  • Card Draw Example 🃏: If A is 'drawing a King' and B is 'drawing a Heart', then A−BA - B is 'drawing a King that is not a Heart', which are the Kings of Spades, Clubs, and Diamonds.

2. Mutually Exclusive Events

Two or more events are mutually exclusive (or disjoint) if they cannot happen at the same time. The occurrence of one event completely rules out the occurrence of the other.

  • In Set Notation: Events A and B are mutually exclusive if their intersection is the empty set: A∩B=∅A \cap B = \emptyset.
  • Venn Diagram: Mutually exclusive events are represented by two circles that do not overlap.
  • Example: In rolling a die, the event A = 'getting an even number' ({2, 4, 6}) and the event B = 'getting an odd number' ({1, 3, 5}) are mutually exclusive. A single roll cannot be both even and odd.
  • Non-Example: The events 'drawing a King' and 'drawing a Heart' from a deck are not mutually exclusive because you can draw the King of Hearts, which satisfies both events.

3. Exhaustive Events

A set of events is exhaustive if their union makes up the entire sample space. This means that when the experiment is performed, at least one of the events in the set is guaranteed to occur.

  • In Set Notation: Events E1_1, E2_2, …, En_n are exhaustive if E1∪E2∪⋯∪En=SE_1 \cup E_2 \cup \dots \cup E_n = S.
  • Example: In rolling a die, the event A = 'getting an even number' ({2, 4, 6}) and event B = 'getting an odd number' ({1, 3, 5}) are exhaustive because their union A∪B={1,2,3,4,5,6}A \cup B = \{1, 2, 3, 4, 5, 6\}, which is the entire sample space SS.

Mutually Exclusive and Exhaustive Events

This is a very important combination. A set of events is mutually exclusive and exhaustive if they do not overlap and they cover all possible outcomes.

  • Example: In rolling a die, the simple events {1}, {2}, {3}, {4}, {5}, and {6} are mutually exclusive and exhaustive.
  • Example: The events 'getting an even number' and 'getting an odd number' are also mutually exclusive and exhaustive.

4. Key Probability Formulas

Based on the algebra of events, we can establish several fundamental rules for calculating probabilities. Let P(A) denote the probability of event A.

Probability of the Complement (Not A)

The probability that an event A will not occur is 1 minus the probability that it will occur. P(A′)=1−P(A)P(A') = 1 - P(A)

Addition Rule for Probability (A or B)

This formula calculates the probability of at least one of two events occurring. We add their individual probabilities and subtract the probability of their intersection to avoid double-counting the overlap. P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Rule for Mutually Exclusive Events

If two events, A and B, are mutually exclusive, they cannot happen at the same time, so P(A∩B)=0P(A \cap B) = 0. The addition rule simplifies: P(A∪B)=P(A)+P(B)(if A and B are mutually exclusive)P(A \cup B) = P(A) + P(B) \quad (\text{if A and B are mutually exclusive})

Rule for 'A but not B'

The probability of event A happening but event B not happening can be found by taking the probability of A and subtracting the probability of their intersection. P(A−B)=P(A)−P(A∩B)P(A - B) = P(A) - P(A \cap B)

Addition Rule for Three Events

For three events, the rule extends to include all pairwise intersections and the triple intersection. P(A∪B∪C)=P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)P(A \cup B \cup C) = P(A)+P(B)+P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)