Question

Two dice are rolled. Let $A$, $B$ and $C$ be the events of getting a sum $2$, sum $3$ and a sum $4$, respectively.
$(i)$Is the event $A$ simple? $(ii)$Is event $B$ simple?
$(iii)$Is event C compound? $(iv)$ Are events $A$ and $B$ mutually exclusive?

Answer 1

Hint: In this question we have to find the probability of events when two dice are thrown together. We will first write the event space for the probability $A$, $B$ and $C$ respectively and then check each case one by one and write the required conclusion.

Complete step by step answer:
We know that $A$, $B$ and $C$ are the events of getting a sum $2$, sum $3$ and a sum $4$respectively.
Now a dice has $6$ values which are from $1$ to $6$.
Now consider $A$, which is the probability of getting the sum of $2$.
The sum of $2$ is only possible when the face of each dice has a value of $1$ therefore, the event space for $A$ is:
$\Rightarrow A=\left\{ \left( 1,1 \right) \right\}$
Now consider $B$, which is the probability of getting the sum of $3$.
The sum of $3$ is possible when the face of either dice has a value of $1$ and the second one has a value $2$, therefore, the event space for $B$ is:
$\Rightarrow B=\left\{ \left( 1,2 \right),\left( 2,1 \right) \right\}$
Now consider $C$, which is the probability of getting the sum of $4$.
The sum of $4$ is possible when the face of either dice has a value of $1$ and the second one has a value $3$ or the face of either dice has a value of $2$, therefore, the event space for $B$ is:
$\Rightarrow C=\left\{ \left( 1,3 \right),\left( 3,1 \right),\left( 2,2 \right) \right\}$
Now consider $(i)$, is event $A$ simple?
An event is considered to be simple if the event space has only $1$ correct answer, since the event space of $A$ contains only $1$ probable event, it is a simple event.
consider $(ii)$, is event $B$ simple?
Since event $B$ has more than one event in the event space, by definition of a simple event, it is not a simple event.
consider $(iii)$, is the event $C$ compound?
An event is considered to be compound if there are more than $1$ outcomes for that probability, since event $C$ has a total of $3$ outcomes, it is a compound event.
consider $(iv)$, Are events $A$ and $B$ mutually exclusive?
$2$ events are considered to be mutually exclusive when the intersection of their event space is a null set, mathematically it can be written as:
$X$ and $Y$ are mutually exclusive if $X\cap Y=\phi $, where $\phi $ is a null set.
Now consider $A\cap B$, since there are no common outcomes in the intersection, we can write it as:
$\Rightarrow A\cap B=\phi $ therefore, we can conclude that the events $A$ and $B$ are mutually exclusive.

Note:
It is to be remembered that when two events are mutually exclusive, the occurrence of one event does not affect another event. When two events are mutually exclusive, it is impossible for them to happen at the same time. It is a general notation for $\phi $, which is called as the null set which means it has no elements, it can be represented as $\phi =\{\}$.