Question

1. a) A die is rolled. Let A be the event that "die shows 4" and B be the event that "die shows an even number". Check whether
( ii) A and B are exhaustive.​

Answer 1

SOLUTION

GIVEN

A die is rolled. Let A be the event that "die shows 4" and B be the event that "die shows an even number".

TO CHECK

Whether A and B are exhaustive.

CONCEPT TO BE IMPLEMENTED

Two events A and B are said to be exhaustive if P(A) + P(B) = 1

EVALUATION

Here it is given that a die is rolled

Then the possible outcomes are

{1,2,3,4,5,6}

So the total number of possible outcomes = 6

Now A be the event that "die shows 4"

Then possible outcomes for the event A is {4}

Then the total number of possible outcomes for the event A is 1

$\displaystyle \sf{P(A) = \frac{1}{6} }$

Now B be the event that "die shows an even number".

So possible outcomes for the event B are {2,4,6}

So the total number of possible outcomes for the event B are 3

$\displaystyle \sf{P(B) = \frac{3}{6} = \frac{1}{2} }$

Now

$\displaystyle \sf{P(A) + P(B) }$

$\displaystyle \sf{ = \frac{1}{6} + \frac{1}{2} }$

$\displaystyle \sf{ = \frac{1 + 3}{6} }$

$\displaystyle \sf{ = \frac{4}{6} }$

$\displaystyle \sf{ = \frac{2}{3} \ne \: 1 }$

Thus A and B are not exhaustive

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Answer 2

ANSWER

IT IS NOT EXHAUSTIVE

STEP BY STEP EXPLANATION

A=4

B=an eben number{2,4,6}

SAMPLE SPACE ={1,2,3,4,5,6}

IF IT IS EXAUSTIVE ,A UNION B=S

A UNION B={2,4,6}

THEREFORE

A UNION B IS NOT EQUAL TO SAMPLE SPACE