Question

A number is selected at random from integers 1 to 100. Find the probability that it is
(i) a perfect square (ii) not a perfect cube.
A.\[\left( i \right)\dfrac{1}{{20}}{\text{ }}\left( {ii} \right)\dfrac{{14}}{{25}}\]
B.\[\left( i \right)\dfrac{1}{{15}}{\text{ }}\left( {ii} \right)\dfrac{{14}}{{25}}\]
C.\[\left( i \right)\dfrac{1}{{10}}{\text{ }}\left( {ii} \right)\dfrac{{24}}{{25}}\]
D.None of these

Answer 1

Hint:
Probability means possibility. It is a branch of mathematics that deals with the occurrence of events. It is used to predict how likely events are to happen. To find the probability of a single event to occur, first, we should know the total number of possible outcomes.
Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.
Random experiment is an experiment where we know the set of all possible outcomes but find it impossible to predict one at any particular execution.
Sample space is defined as the set of all possible outcomes of a random experiment. Example: Tossing a head, Sample Space(S) = {H, T}. The probability of all the events in a sample space adds up to 1.
Event is a subset of the sample space i.e. a set of outcomes of the random experiment.
\[{\text{Probability of an event}} = \dfrac{{{\text{Number of occurence of event A in S}}}}{{{\text{Total number of cases in S}}}}{\text{ }}\]
\[ = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}\]

Complete step by step solution:
Sample space = \[S = \left\{ {1,2,3, \ldots ,100} \right\}\]
\[\therefore {\text{Number of elements in S}} = n\left( S \right) = 100\]
$\left( i \right)$ Let A denotes that number in a perfect square.
\[\therefore A = \left\{ {1,4,9,16,25,36,49,64,81,100} \right\}\]
\[\therefore {\text{n}}\left( A \right) = {\text{Number of elements in A }} = {\text{ 10}}\]
\[\therefore {\text{Required Probability}} = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}\]
\[ = \dfrac{{10}}{{100}}\]
\[ = \dfrac{1}{{10}}\]
$\left( {ii} \right)$ Let B denote a perfect cube number.
\[\therefore B = \left\{ {1,8,27,64} \right\}\]
\[n\left( B \right) = 4\]
\[\therefore {\text{Probability that a number is a perfect cube = }}\dfrac{{n\left( B \right)}}{{n\left( S \right)}}\]
$ = \dfrac{4}{{100}}$
$ = \dfrac{1}{{25}}$
\[{\text{Now, probability that number is not a perfect cube = 1}} - \left( {{\text{probability that a number is a perfect cube}}} \right)\]\[ = 1 - \dfrac{1}{{25}}\]
\[ = \dfrac{{24}}{{25}}\]
∴ Option (C) is correct

Note:
There are some main rules associated with basic probability:
For any event A, 0≤P(A)≤1.
The sum of the probabilities of all possible outcomes is 1.
P(not A)=1-P(A)
P(A or B)=P(event A occurs or event B occurs or both occur)
P(A and B)=P(both event A and event B occurs)
The general Addition Rule: P(A or B) = P(A) + P(B) – P(A and B)