Probability Formulas and Combinatorics Reference

Posted on Aug 5, 2026 in Mathematics

Probability Formulas and Counting Techniques

1. Basic Probability Rules (7.2)

  • Probability of an Event: P(A) = (Number of favorable outcomes) / (Total number of outcomes) = n(A) / n(E)

  • Probability Range: 0 ≤ P(A) ≤ 1

    • P(A) = 0 means the event is impossible.

    • P(A) = 1 means the event is certain to occur.

  • Complementary Events (Not A): P(A’) = 1 – P(A) OR P(A) + P(A’) = 1

  • General Addition Formula (For any two events A and B): P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

  • Mutually Exclusive Events: If A and B cannot happen at the same time, then P(A ∩ B) = 0. Therefore, P(A ∪ B) = P(A) + P(B).

2. Conditional Probability (7.3)

  • Conditional Probability Formula (Probability of A given B): P(A | B) = P(A ∩ B) / P(B) [where P(B) > 0]

  • Multiplication Rule: P(A ∩ B) = P(A | B) * P(B)

  • Law of Total Probability: P(A) = [P(A | B) * P(B)] + [P(A | B’) * P(B’)]

3. Independence (7.4)

Two events A and B are independent if one occurring does not change the likelihood of the other.

  • Test for Independence (If any one statement is true, they are all true):

    1. P(A ∩ B) = P(A) * P(B)

    2. P(A | B) = P(A)

    3. P(B | A) = P(B)

Note: If two events with non-zero probabilities are mutually exclusive, they can never be independent.

4. Visual and Structural Tools

A. Probability Table and Karnaugh Map

Useful for 2-event problems (A and B). All internal entries sum to row/column totals, and all totals sum to 1.

 BNot B (B’)Total
AP(A ∩ B)P(A ∩ B’)P(A)
Not A (A’)P(A’ ∩ B)P(A’ ∩ B’)P(A’)
TotalP(B)P(B’)1

B. Useful Venn Diagram Logic

  • P(Only A): P(A ∩ B’) = P(A) – P(A ∩ B)

  • P(Neither A nor B): P((A ∪ B)’) = 1 – P(A ∪ B)

C. Tree Diagrams

  • Along a branch: Multiply probabilities together.

  • Across different paths: Add probabilities together.

5. Counting Techniques and Combinatorics (7.5)

  • Addition Principle: Selecting between separate, mutually exclusive choices: Total = m + n

  • Multiplication Principle: Sequential multi-step choices: Total = m * n

  • Factorial Notation: n! = n * (n – 1) * (n – 2) * … * 1 [Note: 0! = 1]

Permutations vs. Combinations

  • Permutations (nPr) — Order Matters: Arrangements in a line, specific roles, or positions. Formula: nPr = n! / (n – r)!

  • Combinations (nCr) — Order Does Not Matter: Selecting a group, committee, or hand of cards. Formula: nCr = n! / [r! * (n – r)!]

6. Essential Problem-Solving Strategies (7.5)

  1. “At Least 1” Rule: P(At least 1) = 1 – P(None)

  2. Group / Glue Method (Items must stay together):

    • Treat the tied items as 1 single block.

    • Calculate arrangements of all items including the block.

    • Multiply by internal arrangements inside the block (k!).

  3. Separation / Gap Method (Items must NOT be adjacent):

    • Arrange all unrestricted items first.

    • Insert restricted items into the gaps created between and around the arranged items.

  4. Arrangements in a Circle: Number of ways to seat n distinct items in a circle = (n – 1)!

  5. Permutations with Identical Items: Total arrangements = n! / (p! * q! * …) where p, q, etc., are the counts of identical items.