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What is addition rule for probabilities?

The Addition Rule for probabilities is a fundamental concept in probability theory that provides a method for calculating the probability of the occurrence of either one event or another related event. It applies to situations where events are mutually exclusive or mutually exclusive and exhaustive.

There are two variations of the Addition Rule:

1. Addition Rule for Mutually Exclusive Events:

This rule applies when two events cannot occur simultaneously. The probability of either event A or event B occurring is calculated by adding their individual probabilities.

Mathematically, the rule is expressed as:

P(A or B) = P(A) + P(B)

For example, consider rolling a six-sided die. The probability of rolling a 1 or a 2 can be calculated by adding the individual probabilities:

P(1 or 2) = P(1) + P(2) = 1/6 + 1/6 = 1/3

2. Addition Rule for Mutually Exclusive and Exhaustive Events:

This rule applies when events are mutually exclusive and collectively cover all possible outcomes. In this case, the sum of the probabilities of all possible events is equal to 1.

Mathematically, the rule is expressed as:

P(A or B or C or …) = P(A) + P(B) + P(C) + …

For example, in flipping a fair coin, the probability of getting heads (H) or tails (T) is calculated as:

P(H or T) = P(H) + P(T) = 1/2 + 1/2 = 1

It’s important to note that the Addition Rule assumes that events are independent, meaning that the occurrence of one event does not affect the probability of the other. If events are not independent, more complex probability rules, such as the Multiplication Rule or conditional probabilities, may need to be applied.

The Addition Rule is a useful tool for calculating probabilities when dealing with mutually exclusive or collectively exhaustive events. It enables the assessment of the likelihood of various outcomes and plays a fundamental role in probability calculations and decision-making.