Addition Rule for Probabilities

What is the Addition Rule for Probabilities?

Given multiple events, the addition rule for probabilities is used to compute the probability that at least one of the events happens. Probability can be defined as the branch of mathematics that quantifies the certainty or uncertainty of an event or a set of events.

Related Concepts

Before understanding the addition rule, it is important to understand a few simple concepts:

• Sample space: It is the set of all possible events. For example, when flipping a coin, the sample space is {Heads, Tails} because heads and tails are all the possible outcomes.
• Event: In probability, an event is defined as a particular outcome. For example, flipping a coin and getting heads is an event.
• Mutually exclusive events: They are events such that if one occurs, the other cannot occur. Again, in the coin example, if we get heads, we cannot get tails. Hence, the two are mutually exclusive events.
• Mutually exhaustive events: Events that together encompass the entire sample space. In the case of flipping a coin, getting heads and getting tails are mutually exhaustive as the entire sample space is {Heads, Tails}.
• Independent events: Events that occur independently of each other. For example, when flipping two coins, the outcome of the second coin is independent of the outcome of the first coin.

The formula to compute the probability of two events A and B is given by:

Where:

• P(A ∪ B) – Probability that either A or B happens
• P(A) – Probability of Event A
• P(B) – Probability of Event B
• P(A ∩ B) – Probability of A and B happening together

The following Venn diagram illustrates how and why the formula works:

As shown above, we subtract the P(AB) term because it would be counted twice when adding P(A) and P(B).

Calculating P(A ∩ B)

The probability of events A and B both happening – P(A ∩ B) – can be easily calculated if the events are independent of each other by multiplying the two probabilities P(A) and P(B) as shown below:

If A and B are independent events, then:

If events A and B are not independent of each other, the probability can be inferred from the nature of the events, or it is otherwise difficult to determine.

Mutually Exclusive Events

In case of mutually exclusive events, the probability of both events occurring at once is zero by definition because if one occurs, the other event cannot. Hence, for mutually exclusive events A and B, there is:

Note the fact that mutually exclusive events are not independent because if both P(A) and P(B) are non-zero probabilities, then P(AB) = P(A) * P(B) cannot be zero. In fact, by their very definition of mutually exclusive events, they depend on the other event not occurring. The diagram below illustrates the concept:

Numerical Example

Let’s move on to a numerical example that illustrates the concept. Assume two independent events, A and B. Let P(A) = 0.6 and P(B) = 0.4. Then P(A ∪ B) is given by:

• P(A) = 0.6
• P(B) = 0.4

P(A ∩ B) = P(A) * P(B) = 0.6 * 0.4 = 0.24

P(A ∪ B) = P(A) + P(B) – P(AB) = 0.6 + 0.4 – 0.24 = 0.76

Hence, P(A ∪ B) is 76%.

Derived Rules

The addition rule for probabilities yields some other rules that can be used to calculate other probabilities.

Mutually Exclusive Events

For mutually exclusive events, the joint probability P(A ∪ B) = 0. Hence, we get:

Probability for Exactly One of Two Events

The probability of exactly one of two events can be calculated simply by modifying the addition rule as follows:

More Resources

CFI is the official provider of the global Business Intelligence & Data Analyst (BIDA)® certification program, designed to help anyone become a world-class financial analyst. To keep advancing your career, the additional CFI resources below will be useful:

• Dependent Events vs Independent Events
• Game Theory
• Quantitative Analysis
• Total Probability Rule
• See all data science resources