**What is the Addition Rule for Probabilities, and How Does It Work?**

Two different formulas are described by the addition rule for probabilities. The first formula represents the likelihood that any of two mutually exclusive events will occur, and the second formula describes the probability that two occurrences that are not mutually exclusive will occur.

The first formula summarizes the probability associated with both events. The second formula is the total of the probabilities of both events minus the likelihood that both events will occur.

### The Formulas for the Addition Rules for Probabilities Is

Mathematically, the probability of two mutually exclusive events is denoted by:

$P(YorZ)=P(Y)+P(Z)$

Mathematically, the probability of two non-mutually exclusive events is denoted by:

$P(YorZ)=P(Y)+P(Z)−P(YandZ)$

**What Does It Tell You About the Probabilities If You Add them Together?**

Consider a die with six sides and the odds of rolling either a three or a six to illustrate the first rule in the addition rule for probabilities. Because the odds of rolling a three are one in six and the odds of rolling a six are also one in six, the odds of rolling either a three or a six are as follows: 1/6 + 1/6 = 2/6 = 1/3

Consider a classroom with nine male and 11 female students to demonstrate the second rule. At the end of the semester, the grade of B is awarded to five girls and four boys. If a student is chosen randomly, what are the chances that the student will have a B grade point average or that the student will be a female? The odds of picking a girl or a B student are as follows given that the odds of picking a girl are 11/20, the odds of picking a B student are 9/20, and the odds of picking a girl who is also a B student are 5/20. Given these odds, the chances of picking a girl or a B student are as follows:

11/20 + 9/20 – 5/20 =15/20 = 3/4

The two regulations can be consolidated into a single rule, the second one. This is because, under the first scenario, the probability of two incompatible events occurring simultaneously is 0. It is impossible to roll both a three and a six on the same roll of a single die, as demonstrated by the example with the die. Therefore, the two occurrences cannot occur simultaneously.

**Exclusiveness for Both Parties**

A statistical concept that describes two or more events that cannot occur simultaneously is called “mutually exclusive.” It is a phrase frequently used to explain a scenario in which one event takes precedence over the other. Consider the act of rolling dice as a simple illustration of this concept. On a single die, getting a roll of b five and a three simultaneously is impossible. In addition, the outcome of the first roll having a result of three has no bearing on whether or not the second roll would have a result of five. There is no correlation between any two rolls of a dice.

**Conclusion**

- The addition rule for probabilities is comprised of two separate rules or formulas. The first rule takes into account the occurrence of two events that are incompatible with one another. In comparison, the second rule considers the occurrence of two compatible events.
- The fact that the two events in question are not mutually exclusive indicates some overlap between them. The formula accounts for this overlap by deducting the probability of the overlap, denoted by P(Y and Z), from the total probability of Y and Z.
- In principle, the first version of the rule can be understood as a particular instance of the second form.