>Rules of Probability>What Is the Addition Rule of Probability?

What Is the Addition Rule of Probability?

You use the addition rule when there are several results that match up with the outcome you’re looking for. In that case, you add the probability of each of these outcomes together. If these outcomes have elements in common, and are not disjoint, remember to subtract their intersection. If you forget, you’ll end up counting the shared elements twice:

P (A \cup B) = P (A) + P (B) - P (A \cap B) .

When $A$ and $B$ are disjoint events:

\begin{array}{l} P (A \cap B) & = 0 and \\ P (A \cup B) & = P (A) + P (B) \end{array}

\begin{array}{l} P (A \cap B) & = 0 and \\ P (A \cup B) & = P (A) + P (B) \end{array}

Here, you don’t have to subtract the intersection because it’s empty and is therefore

0

Example 1

Disjoint Events
You throw a strange die with seven differently shaped sides. The probability of getting the different sides is shown in the table below.

$x_{i}$ $P (X = x_{i})$
$1$ $(\frac{1}{19})$
$2$ $(\frac{3}{19})$
$3$ $(\frac{2}{19})$
$4$ $(\frac{7}{19})$
$5$ $(\frac{3}{19})$
$6$ $(\frac{1}{19})$
$7$ $(\frac{2}{19})$

$x_{i}$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
$P (X = x_{i})$ $(\frac{1}{19})$ $(\frac{3}{19})$ $(\frac{2}{19})$ $(\frac{7}{19})$ $(\frac{3}{19})$ $(\frac{1}{19})$ $(\frac{2}{19})$

What is the probability of getting a maximum of three dots?

As the number of dots on one side doesn’t affect the number of dots on a different side, this experiment is disjoint:

\begin{array}{l} P (X \leq 3) \\ = P (\frac{1}{19}) + P (\frac{3}{19}) + P (\frac{2}{19}) \\ = \frac{6}{19} \end{array}

P (X \leq 3) = P (\frac{1}{19}) + P (\frac{3}{19}) + P (\frac{2}{19}) = \frac{6}{19}

Example 2

Non-Disjoint Events

You have a group of 20 friends who play different sports. 15 are skiers, 12 play football and 2 don’t play any sports. What is the probability that one random person in the group is a skier, plays football, or both?

Call the outcome that a friend is a skier $S$ , and the outcome that a friend is a footballer $F$ . First we need the intersection between people who are skiers and people who play football:

S \cap F = 15 + 12 + 2 - 20 = 9

Then you find

\begin{array}{l} P (S \cup F) & = P (S) + P (F) \\ - P (S \cap F) \\ = \frac{15}{20} + \frac{12}{20} - \frac{9}{20} \\ = \frac{18}{20} \\ = \frac{9}{10} \end{array}

\begin{array}{l} P (S \cup F) & = P (S) + P (F) - P (S \cap F) \\ = \frac{15}{20} + \frac{12}{20} - \frac{9}{20} \\ = \frac{18}{20} \\ = \frac{9}{10} . \end{array}

This tells us that

90

% of the group are skiers, or play football, or both.

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$x_{i}$	$P (X = x_{i})$

$1$	$(\frac{1}{19})$

$2$	$(\frac{3}{19})$

$3$	$(\frac{2}{19})$

$4$	$(\frac{7}{19})$

$5$	$(\frac{3}{19})$

$6$	$(\frac{1}{19})$

$7$	$(\frac{2}{19})$