Addition Rule For Probabilities
The addition rule for probabilities describes two formulas, one for the probability for either of two mutually exclusive events happening and the other for the probability of two non-mutually exclusive events happening.The first formula is just the sum of the probabilities of the two events. The second formula is the sum of the probabilities of the two events minus the probability that both will occur.
Probability Addition Rule
Definition
The Probability Addition Rule is a fundamental theorem in probability theory used to calculate the combined probability of two events occurring. It includes two formulas: one for mutually exclusive events and another for non-mutually exclusive events.
Origin
The concept of the Probability Addition Rule dates back to the 17th century when mathematicians like Pascal and Fermat began systematically studying probability theory. Over time, these basic theorems were further developed and refined, forming the foundation of modern probability theory.
Categories and Characteristics
1. Mutually Exclusive Events: If two events A and B are mutually exclusive (i.e., they cannot occur simultaneously), their combined probability P(A or B) is the sum of their individual probabilities, i.e., P(A or B) = P(A) + P(B).
2. Non-Mutually Exclusive Events: If two events A and B are not mutually exclusive (i.e., they can occur simultaneously), their combined probability P(A or B) is the sum of their individual probabilities minus the probability of both events occurring together, i.e., P(A or B) = P(A) + P(B) - P(A and B).
Examples
Example 1: Suppose you have a bag containing red and blue balls. You randomly draw one ball from the bag. Event A is drawing a red ball, and Event B is drawing a blue ball. Since a ball cannot be both red and blue at the same time, events A and B are mutually exclusive. If P(A) = 0.3 and P(B) = 0.5, then P(A or B) = 0.3 + 0.5 = 0.8.
Example 2: Suppose you have a bag containing red, blue, and green balls. You randomly draw one ball from the bag. Event A is drawing a red ball, and Event B is drawing a blue ball. If P(A) = 0.3, P(B) = 0.5, and P(A and B) = 0.1 (the probability of drawing a ball that is both red and blue), then P(A or B) = 0.3 + 0.5 - 0.1 = 0.7.
Common Questions
1. How to distinguish between mutually exclusive and non-mutually exclusive events? Mutually exclusive events cannot occur simultaneously, while non-mutually exclusive events can occur at the same time.
2. Why subtract P(A and B)? When calculating the combined probability of non-mutually exclusive events, P(A) and P(B) both include P(A and B), so it needs to be subtracted once to avoid double-counting.
