Addition Theorem of Probability: Calculating Probabilities of Mutually Exclusive Events

Understand the addition theorem of probability for calculating the probability of either of two mutually exclusive events occurring. This guide explains the theorem, provides examples, and demonstrates its application in solving probability problems.

The Addition Theorem of Probability

Theorem 1: Mutually Exclusive Events

If events A and B are mutually exclusive (they cannot both happen at the same time), then the probability of A or B occurring is the sum of their individual probabilities:

P(A ∪ B) = P(A) + P(B)

Example 1: Mutually Exclusive Events

Two dice are rolled. What's the probability of getting an even number on the first die or a total sum of 8?

(The solution, showing the calculation of the individual probabilities and their sum, would be included here.)

Theorem 2: Non-Mutually Exclusive Events

If events A and B are not mutually exclusive (they can both happen), then the probability of A or B occurring is:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

We subtract P(A ∩ B) (the probability of both A and B occurring) because it's been double-counted in P(A) and P(B).

Example 1: Non-Mutually Exclusive Events

Two dice are rolled. What's the probability of getting an even number on the first die or a total sum of 8?

(The solution, showing the calculation of P(A), P(B), P(A ∩ B), and the application of the formula would be included here.)

Example 2: Classifying Events

Two dice are rolled. Consider these events:

  • A: Even number on the first die
  • B: Odd number on the first die
  • C: Sum ≤ 5
  • D: Sum > 5 and < 10
  • E: Sum ≥ 10
  • F: Odd number on at least one die

Let's analyze the relationships between these events:

  • A and B are mutually exclusive and exhaustive (one must happen).
  • A and C are not mutually exclusive.
  • C and D are mutually exclusive but not exhaustive.
  • C, D, and E are mutually exclusive and exhaustive.
  • A' and B' are mutually exclusive and exhaustive.
  • A, B, and F are not mutually exclusive.

(The sets representing each event would be listed here.)

Conclusion

The addition theorem of probability provides a way to calculate the probability of the union of events, whether those events are mutually exclusive or not. Understanding these concepts is fundamental to probability theory.