Events: types and operations

An event is any subset of the sample space $S$. We can combine events using set operations to form new events. Because events are sets, all the algebra of sets carries over.

Types of events

• Sure event (or certain event): the whole sample space $S$. It always occurs.
• Impossible event: the empty set $\varnothing$. It never occurs.
• Simple (or elementary) event: a singleton , a set containing exactly one outcome.
• Compound event: an event with more than one outcome.

Examples on rolling a die ($S = \{1, 2, 3, 4, 5, 6\}$):

• Sure event: $\{1, 2, 3, 4, 5, 6\}$.
• Impossible: $\{x \in S : x > 6\} = \varnothing$.
• Simple: $\{4\}$ (the die shows $4$).
• Compound: $\{2, 4, 6\}$ (an even number).

Operations on events

If $A, B$ are events:

• Union $A \cup B$: outcomes in $A$ or $B$ (or both). "Either happens."
• Intersection $A \cap B$: outcomes in both $A$ and $B$. "Both happen."
• Complement $A'$ (or $A^c$): outcomes not in $A$. "$A$ does not happen."
• Difference $A - B = A \cap B'$: outcomes in $A$ but not in $B$.

These have the usual properties (commutative, associative, distributive, De Morgan) you learned in Chapter 1 on sets.

Special relationships

• Mutually exclusive (disjoint): $A \cap B = \varnothing$ , can't both happen.
• Exhaustive: $A_1 \cup A_2 \cup \dots \cup A_n = S$ , at least one must happen.
• Mutually exclusive and exhaustive: a partition of $S$.

For instance, the events "even" and "odd" on a die are mutually exclusive and exhaustive.

Worked examples

Example 1. Roll a die. Let $A =$ "even", $B =$ "less than $4$". Find $A \cup B, A \cap B, A', B - A$.

$A = \{2, 4, 6\}$, $B = \{1, 2, 3\}$.

$A \cup B = \{1, 2, 3, 4, 6\}$. $A \cap B = \{2\}$. $A' = \{1, 3, 5\}$. $B - A = \{1, 3\}$.

Example 2. Toss three coins. Let $A =$ "at least two heads", $B =$ "first coin is heads". List $A, B, A \cap B$.

$A = \{HHH, HHT, HTH, THH\}$. $B = \{HHH, HHT, HTH, HTT\}$.

$A \cap B = \{HHH, HHT, HTH\}$.

Example 3. Are $A = \{1, 2\}$ and $B = \{3, 4\}$ on a die mutually exclusive? Exhaustive?

$A \cap B = \varnothing$ , yes, mutually exclusive. $A \cup B = \{1, 2, 3, 4\} \ne S$ , not exhaustive.

Example 4. Two dice rolled. Event "doubles" and event "sum > $10$". List each, find their intersection.

"Doubles" $= \{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\}$. "Sum > 10" $= \{(5,6), (6,5), (6,6)\}$. Intersection $= \{(6,6)\}$.

Example 5. Three coins. $A =$ "exactly one head", $B =$ "at most one head". Relate $A$ and $B$.

$A = \{HTT, THT, TTH\}$. $B = \{TTT, HTT, THT, TTH\}$. So $A \subset B$.

Try it yourself

1. Roll a die. $A =$ "multiple of $3$", $B =$ "less than $5$". List $A, B, A \cap B, A \cup B$.
2. Two coins tossed. $A =$ "at least one head". $A'$?
3. Roll two dice. List the event "sum is $9$".
4. Are "head on first toss" and "head on second toss" mutually exclusive in two coin tosses?
5. Three coin tosses. Events: "exactly $1$ head", "exactly $2$ heads", "exactly $3$ heads", "no head". Are they mutually exclusive? Exhaustive?
6. Draw a card. $A =$ "diamond", $B =$ "face card". $A \cap B$?
7. Two dice. "At least one shows $6$". List or count.
8. A coin tossed $4$ times. Event "all same". List.
9. Roll a die. "Number is prime". List.
10. From $\{1, 2, \dots, 10\}$ a number is chosen. "Multiple of $2$" and "multiple of $3$". Find intersection and union.
11. Two dice rolled. Event "sum is even". $|E|$?
12. Three coins tossed. List the event "more heads than tails".

Pitfalls / Tricks

• Always list events explicitly when the sample space is small , avoids mistakes.
• "At least one" is the complement of "none".
• Mutually exclusive: $\cap = \varnothing$. Exhaustive: $\cup = S$.
• Insight. Events are sets. Every identity from set theory translates directly into a probability identity (after applying $P$).