Explore: Compound Events Solutions

Compound Events Solutions

A compound event in probability is made up of two or more simple events that happen together or in sequence.

Mutually Exclusive Events

Note

This is always true because there are no overlapping elements.

If A and B are mutually exclusive events, then the union , ⋃, of the events is:
$P (A or B) = P (A) + P (B)$
or
$P (A \cup B) = P (A) + P (B)$

Note

Notice the word “or” can be replaced by the union symbol ⋃.

Inclusive events

Inclusive events occur when parts of a set overlap or intersect with another set. 
To prevent double-counting , subtract the intersection, ⋂, from the sum of A and B: 
Ask yourself, “What elements do A AND B have in common ?”

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

Sort the set, ${1, 2, . . ., 14}$ , into factors of 18 and multiples of 4. Then find the probabilities.

$P (factor of 18 \cup multi . of 4) = \frac{5}{14} + \frac{3}{14} = \frac{8}{14} = \frac{4}{7}$

$P (multiple of 4 or not a factor of 18)$
$= \frac{3}{14} + \frac{6}{14} = \frac{9}{14}$

Note

Because there are no numbers that are both factors of 18 and multiples of 4, the events (or numbers) are mutually exclusive. 

Remember, you need to count how many numbers, or elements, are in each section to find the probability. Do not use the numbers themselves.

Determine probabilities of a standard deck of cards.

$= \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$

$= \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$

$= \frac{12}{52} + \frac{26}{52} - \frac{6}{52} = \frac{32}{52} = \frac{8}{13}$

$P (red \cap face card) = \frac{6}{52} = \frac{3}{26}$

$P (black ten \cup red seven) = \frac{2}{52} + \frac{2}{52} = \frac{4}{52} = \frac{1}{13}$

Note

Problems A and E represent mutually exclusive events.