Events and their types

What you will be able to do

• write an event as a subset of the sample space and say when it happens;
• tell simple, compound, certain and impossible events apart;
• decide whether two events are mutually exclusive, exhaustive or both.

From a question to a set

Two dice are thrown. "The total is $7$." Which outcomes does this sentence select? The pairs $(1, 6)$, $(2, 5)$, $(3, 4)$, $(4, 3)$, $(5, 2)$ and $(6, 1)$. A sentence about the result has become a set of outcomes. That set is an event.

An event is a subset of the sample space. We say the event happens, or occurs, when the outcome of the experiment is one of the members of the set.

Kinds of event

• A simple event (also called an elementary event) has exactly one outcome. "The total is $12$" is simple, because only $(6, 6)$ gives it.
• A compound event has more than one outcome. "The total is $7$" is compound.
• The sure event (or certain event) is the whole sample space $S$. It always happens.
• The impossible event is the empty set $\varnothing$. It never happens. "The total is $13$" is impossible with two dice.

Every compound event is made of simple events. In a fair experiment, the probability of a compound event comes from counting the simple events in it.

From words to sets

The words of a question hide the set. Learn these:

• At least $k$ means $k$ or more. "At least two heads" in three tosses is $HHH, HHT, HTH, THH$.
• At most $k$ means $k$ or fewer.
• Exactly $k$ means only $k$.
• More than $k$ does not include $k$. Less than $k$ does not include $k$.

Two events together

Let $A$ and $B$ be events of the same experiment.

• $A$ and $B$ are mutually exclusive if they cannot happen together. This means $A \cap B = \varnothing$, which says that no outcome is in both. A throw of a die cannot show both "even" and "$5$".
• $A$ and $B$ are exhaustive if one of them must happen. This means $A \cup B = S$. The events "at most $4$" and "at least $3$" are exhaustive for a die.
• If both are true, the events are mutually exclusive and exhaustive. Then each is the complement of the other. "The total is even" and "the total is odd" are such a pair.

Several events can be mutually exclusive and exhaustive together, such as $E_1, E_2, \dots, E_k$, when each pair is mutually exclusive and their union is $S$. Such a family cuts the sample space into pieces, and it will be needed in the Class 12 chapter on total probability and Bayes' theorem.

Do not mix up two words. Mutually exclusive means the events cannot happen together. Independent is a different idea: it means that knowing one happened does not change the chance of the other. You will meet it in the Class 12 chapter on conditional probability. Two events that are mutually exclusive, each with a chance above zero, are never independent.

A worked example with a die

A die is thrown. Let $A$ be "the number is a multiple of $3$", $B$ be "the number is a power of $2$", $C$ be "the number is odd".

1. List the events. $A = \{3, 6\}$, $B = \{1, 2, 4\}$, $C = \{1, 3, 5\}$.
2. $A$ and $B$. They have no common member, so they are mutually exclusive. Their union is $\{1, 2, 3, 4, 6\}$, which is not $S$ because $5$ is missing. So they are not exhaustive.
3. $B$ and $C$. The number $1$ is in both. They are not mutually exclusive.
4. $C$ and $\{2, 4, 6\}$. They are mutually exclusive, and their union is $S$. They are mutually exclusive and exhaustive.

Practise with new numbers

Try it in Python

Python can test the two definitions. The program builds events on two dice as sets. Mutually exclusive means an empty overlap, and exhaustive means that the union is the whole sample space.

Common mistakes

• Saying "exclusive" when you mean "independent". Exclusive events cannot happen together. They depend on each other completely.
• Forgetting that exhaustive needs the whole sample space. Two events can be mutually exclusive and still leave some outcomes out.
• Taking "at most $k$" as "less than $k$". At most $k$ includes $k$.
• Calling an event with one outcome compound. An event with exactly one outcome is simple.

Remember these

• An event is a subset of $S$. It happens when the outcome belongs to it.
• Simple: one outcome. Compound: more. Sure: $S$. Impossible: $\varnothing$.
• Mutually exclusive: $A \cap B = \varnothing$. Exhaustive: $A \cup B = S$.
• Both together: each event is the complement of the other.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Two coins are tossed. Write the event "at least one head" as a set.

2. A die is thrown. Are the events "the number is less than $3$" and "the number is more than $4$" mutually exclusive? Are they exhaustive?

3. Two dice are thrown. Is "the total is $13$" an event? What is it called?

4. A card is drawn from a pack. Give a pair of events that are mutually exclusive and exhaustive.

5. Are mutually exclusive events with a chance above zero independent?

References

• NCERT, Mathematics for Class XI (earlier edition), Probability, section on events and the algebra of events.
• OpenStax, Introductory Statistics, Chapter 3, section on independent and mutually exclusive events.