Question

A bag contains 30 balls numbered from 1 to 30, one ball is drawn randomly. The probability that the number on the ball is multiple of 5 or 7 is
A.\[\dfrac{1}{2}\]
B.\[\dfrac{1}{3}\]
C.\[\dfrac{2}{3}\]
D.\[\dfrac{1}{4}\]

Answer 1

Hint: Here the given question is based on the concept of probability. We have to find the probability of choosing a ball which has multiple of 5 or 7 by adding the probability of multiple of 5 and multiple of 7. For this, first we need to find the set of numbers which has a multiple of 5 and 7 between 1 to 30 then by using the definition of probability and on further simplification we get the required probability of choosing a card.

Complete step-by-step answer:
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are to happen, using it. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.
The probability formula is defined as the probability of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.
\[Probability \;of \;event\; to\; happen\; P\left( E \right) = \dfrac{{Number\; of\; favourable \;outcomes}}{{Total \;Number\; of \;outcomes}}\]
Consider the given question:
A bag contains 30 balls. One ball are drawn randomly from the from to bag,
Total balls numbered from 1 to 30 \[ = 30\]
 now we have to find
Probability of a drawn ball is multiple of 5.
The numbers which multiple of 5 from 1 to 30 are 5, 10, 15, 20, 25, and 30.
Total number of multiple of 5 from 1 to 30 \[ = 6\]
By the definition of probability
\[ \Rightarrow P\left( {{\text{Multiple of }}5} \right) = \dfrac{{Total\,number\,of\,multiple of \;{\text{5}}}}{{Total{\text{ }}numbered{\text{ }}balls{\text{ }}from{\text{ }}1{\text{ }}\;to{\text{ 30}}}}\]
\[ \Rightarrow P\left( {{\text{Multiple of }}5} \right) = \dfrac{6}{{30}}\]
Probability of drawn ball is multiple of 7
The numbers which multiples of 7 from 1 to 30 are
7, 14, 21, and 28.
Total number of multiples of 7 from 1 to 30\[ = 4\]
By the definition of probability
\[ \Rightarrow P\left( {{\text{Multiple of 7}}} \right) = \dfrac{{Total\,number\,of\,multiple \;of \;7}}{{Total{\text{ }}numbered{\text{ }}balls{\text{ }}from{\text{ }}1{\text{ }}\;to{\text{ 30}}}}\]
\[ \Rightarrow P\left( {{\text{Multiple of 7}}} \right) = \dfrac{4}{{30}}\]
The probability that the number on the ball is a multiple of 5 or 7 is:
\[ \Rightarrow P\left( {{\text{Multiple of }}5} \right) + P\left( {{\text{Multiple of 7}}} \right)\]
\[ \Rightarrow \dfrac{6}{{30}} + \dfrac{4}{{30}}\]
\[ \Rightarrow \dfrac{{6 + 4}}{{30}}\]
\[ \Rightarrow \dfrac{{10}}{{30}}\]
Divide both numerator and denominator by 10, then we get
\[ \Rightarrow \dfrac{1}{3}\]
Hence, the probability that the number on the ball is multiple of 5 or 7 is \[\dfrac{1}{3}\].
Therefore, option (B) is the correct answer.
So, the correct answer is “Option B”.

Note: The probability is a number of possible values, students must know the definition and the basic theorem of probability like addition and multiplication theorem. The word ‘or’ will represents the union of two events ‘and’ the word and represents the intersection of two events. If there is no common term in both events it means it is called independent events.