> ## Documentation Index
> Fetch the complete documentation index at: https://v1-learn.neoartd.my.id/llms.txt
> Use this file to discover all available pages before exploring further.

# The Addition Rules for Probability

Two events are `mutually exclusive` if they cannot occur at the same time (i.e. they have no outcomes in common).

> Dua peristiwa disebut `mutually exclusive` jika mereka tidak dapat terjadi pada saat yang sama (yaitu mereka tidak memiliki hasil yang sama).

![Mutually Exclusive Events](https://i.ibb.co/qnRCKxn/Screenshot-2024-05-01-164644.png)

## Addition Rule 1

<Card>
  When two events A and B are mutually exclusive, the probability that A or B will occur is

  <br />

  $~~~~~ P(A \text{ or } B) = P(A) + P(B) $
</Card>

### Addition Rule 1 - Example

<Card>
  At a political rally, there are 20
  Republicans (R), 13 Democrats (D),
  and 6 Independents (I). If a person is
  selected, find the probability that he or
  she is either a Democrat or an
  Independent.

  <br />

  Solution:

  $P(D \text{ or } I) = P(D) + P(I) = \dfrac{13}{39} + \dfrac{6}{39} = \dfrac{19}{39}$
</Card>

<Card>
  A day of the week is selected at
  random. Find the probability that
  it is a weekend.

  <br />

  Solution:

  $P(\text{Saturday or Sunday}) = P(\text{Saturday}) + P(\text{Sunday}) = \dfrac{1}{7} + \dfrac{1}{7} = \dfrac{2}{7}$
</Card>

## Addition Rule 2

<Card>
  When two events A and B
  are <u>not</u> mutually exclusive, the
  probability that A or B will
  occur is

  <br />

  $~~~~~ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) $

  <br />

  ![Addition Rule 2](https://i.ibb.co/8B0rmxS/Screenshot-2024-05-01-165443.png)
</Card>

### Addition Rule 2 - Example

<Card>
  In a hospital unit there are eight
  nurses and five physicians. Seven
  nurses and three physicians are
  females. If a staff person is selected,
  find the probability that the subject is
  a nurse or a male.

  |   STAFF   | FEMALES | MALES | TOTAL |
  | :-------: | :-----: | :---: | :---: |
  |   NURSE   |    7    |   1   |   8   |
  | PHYSICIAN |    3    |   2   |   5   |
  |   TOTAL   |    10   |   3   |   13  |

  <br />

  Solution:

  $P(\text{nurse or male}) = P(\text{nurse}) + P(\text{male}) + P(\text{male and nurse}) = \dfrac{8}{13} + \dfrac{3}{13} - \dfrac{1}{13} = \dfrac{10}{13}$
</Card>

<Card>
  On New Year's Eve, the probability that a
  person driving while intoxicated is 0.32,
  the probability of a person having a
  driving accident is 0.09, and the
  probability of a person having a driving
  accident while intoxicated is 0.06. What is
  the probability of a person driving while
  intoxicated or having a driving accident?

  <br />

  Solution:

  $P(\text{intoxicated or accident}) = P(\text{intoxicated}) + P(\text{accident}) - P(\text{intoxicated and accident}) = 0.32 + 0.09 - 0.06 = 0.35$
</Card>