The Addition Rule for Probability provides a way to calculate the probability of two outcomes or events that are related to each other. It accounts for situations when one or both of the events overlap, occur at the same time, or have an impact on each other. The rule is based on three main principles:

The first principle of the Addition Rule for Probability states that the probability of one event plus the probability of another event must equal the probability of both events happening at the same time. This can be represented as:

P(Y) + P(Z) = P(Y and Z)

For example, if the probability of event Y occurring is 0.4 and the probability of event Z occurring is 0.3, then the probability of both events happening at the same time would be 0.7.

The second principle of the Addition Rule states that when two events are not mutually exclusive, meaning they have some overlap, then the probability of both events occurring at the same time is the sum of their probabilities minus the probability of them both occurring together. This can be represented as:

P(Y) + P(Z) - P(Y and Z) = P(Y or Z)

In this case, P(Y or Z) represents the probability of either Y or Z occurring.

For example, the probability of event Y occurring is 0.4 and the probability of event Z occurring is 0.3 and the probability of them both occurring together is 0.2. The addition rule would then calculate the probability of either Y or Z occurring as follows:

P(Y) + P(Z) - P(Y and Z)= 0.4 + 0.3 - 0.2 = 0.5

In this situation, the probability of either Y or Z occurring is 0.5.

The third principle of the Addition Rule for Probability states that when two events are mutually exclusive, meaning there is no overlap between them, then the probability of both events occurring at the same time is the sum of their probabilities. This can be represented as:

P(Y) + P(Z) = P(Y or Z)

For example, the probability of event Y occurring is 0.4 and the probability of event Z occurring is 0.3. The addition rule would then calculate the probability of either Y or Z occurring as follows:

P(Y) + P(Z) = 0.4 + 0.3 = 0.7

In this situation, the probability of either Y or Z occurring is 0.7.

The Addition Rule for Probability is an important tool for determining the likelihood of combined events or outcomes and accounting for overlapping probabilities. It can be used to calculate the probability of two simultaneous events or two non-simultaneous events that overlap. By following the three principles of the Addition Rule for Probability and taking into account the probability of overlap, it is possible to accurately calculate the overall probability of two events reaching a desired outcome.