Events, Venn Diagrams and the Addition Rule

Events are sets of outcomes

The sample space is the set of every possible outcome. An event is any collection of those outcomes.

Draw one card from a standard deck of $52$.

Event	Outcomes in it	Count
a heart	the $13$ hearts	$13$
a king	the $4$ kings	$4$
a red card	hearts and diamonds	$26$

Because events are sets, they combine the way sets do. Three words carry the whole vocabulary:

Word	Set name	Symbol	Contains
and	intersection	$A \cap B$	outcomes in both events
or	union	$A \cup B$	outcomes in either event, or both
not	complement	$A^c$ or $\overline{A}$	outcomes not in the event

“A heart and a king” is a single card, the king of hearts. “A heart or a king” is $16$ cards, which the next sections explain.

Picturing events with a Venn diagram

The left circle holds event $A$ and the right circle event $B$. The shaded overlap is $A \cap B$. Everything inside either circle is $A \cup B$, and everything outside both is the complement of $A \cup B$.

Region	Event
the shaded overlap	$A$ and $B$
inside either circle	$A$ or $B$
outside a circle	not that event

The complement

An outcome is either in an event or not in it, never both and never neither. So the two probabilities fill the whole sample space:

$$P(\text{not } A) = 1 - P(A)$$

The complement is often the faster route. “At least one six in four rolls” has many cases; “no sixes in four rolls” has one.

$$P(\text{at least one six}) = 1 - \left(\frac{5}{6}\right)^4 \approx 1 - 0.482 = 0.518$$

Why the addition rule subtracts the overlap

Try to find $P(\text{heart or king})$ by adding:

$$P(\text{heart}) + P(\text{king}) = \frac{13}{52} + \frac{4}{52} = \frac{17}{52}$$

That is one card too many. The king of hearts is a heart and a king, so it was counted once among the hearts and again among the kings.

Subtract it once, and every card is counted exactly one time:

$$P(\text{heart or king}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

In general:

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

The overlap is added twice by the first two terms, so the third term removes one copy. In the Venn diagram, the shaded lens sits inside both circles — exactly the region counted twice.

Mutually exclusive events

Two events are mutually exclusive when they cannot happen together. Their circles do not overlap, so $P(A \cap B) = 0$ and the rule becomes plain addition:

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

Events on one card	Mutually exclusive?	Why
a heart, a spade	yes	no card is both suits
a heart, a king	no	the king of hearts is both
a face card, an ace	yes	an ace is not a face card

Check for an overlap before adding. Plain addition is only safe once you know there is nothing to subtract.

Reading the rule from a table

A survey of $200$ students asks whether they play a sport and whether they are in the band.

	Band	No band	Total
Sport	$18$	$42$	$60$
No sport	$32$	$108$	$140$
Total	$50$	$150$	$200$

$$P(\text{sport or band}) = \frac{60}{200} + \frac{50}{200} - \frac{18}{200} = \frac{92}{200} = 0.46$$

A check straight from the table agrees. The students in neither activity are the $108$ in the bottom-right cell, so the rest, $200 - 108 = 92$, are in at least one.

Worked examples

Common mistakes

Practice problems