Question

A series with three matches between India and South Africa is about to start. Consider the three events associated with the result of the match.
A. SA wins at least two matches
B. SA wins exactly one match
C. SA wins no match

Statement: The three events A, B and C are mutually exclusive and exhaustive events.

• A. True
• B. False

Answer 1

The correct option is A

True

Two events are said to be mutually exclusive if there is no common element in both or $A\cap B=\varphi$.

A set of events $E_1,E_2,E_3.....E_n$ are said to be exhaustive, if $E_1\cup E_2\cup E_3\cup \dots E_n=S$. (sample space)

Let us find out the sample space by denoting South Africa winning as S and India winning as I.

S = {SSS, SSI, SIS, ISS, SII, ISI, IIS, III}

There are eight elements in the sample space S.

Now let's find out the events A, B and C

Event A

A is defined as SA wins at least two matches. From the sample space we can see that there are 4 elements with at least two SA wins in it.

$\Rightarrow$ A = {SSS, SSI, SIS, ISS}

Event B

SA wins exactly one match

$\Rightarrow$ B = {SII, ISI, IIS}

Event C

SA wins no match

C = {III}

Now let's check if the given events A, B and C are mutually exclusive and exhaustive.

$A\cap B=\varphi ,B\cap C=\varphi ,C\cap A=\varphi$

$\Rightarrow$ A, B and C are mutually exclusive

$A\cup B\cup C$ = {SSS, SSI, SIS, ISS, SII, ISI, IIS, III}

= Sample space

$\Rightarrow$ Exhaustive

$\Rightarrow$ Mutually exclusive and exhaustive.