General Addition Rule of Probability

Fundamentals of Probability and the Union of Events

  • The term "or" refers to the union of events, representing all outcomes that satisfy the first condition, the second condition, or both simultaneously.

  • In Venn diagrams, the union encompasses all elements located within either of the specified sets.

Addition Rule for Disjoint Events

  • Disjoint (mutually exclusive) events cannot occur at the same time and share no common elements or overlap.

  • For disjoint events, the probability of either event occurring is the direct sum of their individual probabilities:

    • In a set of 1010 total poker chips with 22 blue chips and 33 green chips:

    • P(blue)=210=20%P(\text{blue}) = \frac{2}{10} = 20\%

    • P(green)=310=30%P(\text{green}) = \frac{3}{10} = 30\%

    • P(blue and green)=0P(\text{blue and green}) = 0

    • P(blue or green)=P(blue)+P(green)=210+310=510=50%P(\text{blue or green}) = P(\text{blue}) + P(\text{green}) = \frac{2}{10} + \frac{3}{10} = \frac{5}{10} = 50\%

Non-Disjoint Events and Double Counting

  • Non-disjoint events have an intersection, meaning certain outcomes satisfy both conditions at once.

  • Adding probabilities directly without adjustment causes overlapping outcomes to be counted twice:

    • Given 1010 chips with 55 red chips and 55 even-numbered chips:

    • P(red)=510=0.5P(\text{red}) = \frac{5}{10} = 0.5

    • P(even)=510=0.5P(\text{even}) = \frac{5}{10} = 0.5

    • Adding these directly yields 0.5+0.5=1.00.5 + 0.5 = 1.0 (100%100\%), which incorrectly implies that every chip is red or even.

    • Chips numbered 22 and 44 are both red and even, resulting in their double counting.

    • The true number of winning chips is 88 out of 1010.

The General Addition Rule

  • The general addition rule accounts for double counting by subtracting the intersection (the overlap) of both events:

    • P(A or B)=P(A)+P(B)−P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

  • Applying the rule to the red and even chip scenario:

    • P(red or even)=P(red)+P(even)−P(red and even)=510+510−210=810P(\text{red or even}) = P(\text{red}) + P(\text{even}) - P(\text{red and even}) = \frac{5}{10} + \frac{5}{10} - \frac{2}{10} = \frac{8}{10}

  • The general addition rule is universally applicable:

    • When events are disjoint, P(A and B)=0P(A \text{ and } B) = 0, simplifying the formula to P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B).