Addition theorem of probability
Addition Theorem of Probability
The addition theorem of probability is a fundamental concept in the field of probability theory. It provides a way to calculate the probability of the union of two or more events. Understanding this theorem is crucial for solving a wide range of probability problems.
Basic Concepts
Before diving into the addition theorem, let's review some basic probability concepts:
- Experiment: A process that leads to one of several possible outcomes.
- Sample Space (S): The set of all possible outcomes of an experiment.
- Event (A, B, ...): A subset of the sample space. An event occurs if the outcome of the experiment is an element of the event.
Probability of an Event
The probability of an event A, denoted by P(A), is a measure of the likelihood that A will occur when the experiment is performed. It satisfies the following properties:
- $0 \leq P(A) \leq 1$
- $P(S) = 1$
- If A and B are mutually exclusive events, then $P(A \cup B) = P(A) + P(B)$
Addition Theorem of Probability
The addition theorem comes in two forms: one for mutually exclusive events and one for non-mutually exclusive events.
Mutually Exclusive Events
Two events A and B are mutually exclusive (or disjoint) if they cannot occur at the same time, i.e., $A \cap B = \emptyset$. For such events, the addition theorem states:
$$ P(A \cup B) = P(A) + P(B) $$
Non-Mutually Exclusive Events
When two events A and B are not mutually exclusive, they can occur together. In this case, the addition theorem is:
$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$
This formula accounts for the fact that the intersection of A and B has been counted twice and must be subtracted once to get the correct probability.
Table of Differences and Important Points
| Aspect | Mutually Exclusive Events | Non-Mutually Exclusive Events |
|---|---|---|
| Definition | $A \cap B = \emptyset$ | $A \cap B \neq \emptyset$ |
| Addition Theorem | $P(A \cup B) = P(A) + P(B)$ | $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ |
| Intersection Probability | $P(A \cap B) = 0$ | $P(A \cap B) > 0$ |
| Example | Flipping a coin (cannot be heads and tails) | Drawing a card (can be a heart and a king) |
Examples
Example 1: Mutually Exclusive Events
Suppose you have a standard deck of 52 cards. Event A is drawing a heart, and Event B is drawing a spade. These events are mutually exclusive because a card cannot be both a heart and a spade.
$$ P(A) = \frac{13}{52} $$ $$ P(B) = \frac{13}{52} $$ $$ P(A \cup B) = P(A) + P(B) = \frac{13}{52} + \frac{13}{52} = \frac{26}{52} = \frac{1}{2} $$
Example 2: Non-Mutually Exclusive Events
Now, let's say Event A is drawing a heart, and Event B is drawing a king. These events are not mutually exclusive because one card (the king of hearts) is both a heart and a king.
$$ P(A) = \frac{13}{52} $$ $$ P(B) = \frac{4}{52} $$ $$ P(A \cap B) = \frac{1}{52} $$ $$ P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13} $$
Conclusion
The addition theorem of probability is a powerful tool for calculating the probability of the union of events. It is essential to distinguish between mutually exclusive and non-mutually exclusive events to apply the correct formula. Understanding and applying this theorem is a key skill for anyone studying probability.