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Thirty identical cards are marked from 1 to 30. If one of the cards is drawn at random, find the probability that it is:
(i) a multiple of 4 or 6
(ii) a multiple of 3 and 5
(iii) a multiple of 3 or 5

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Last updated date: 20th Jun 2024
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Answer
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Hint: According to the basic definition of probability, probability of occurrence of any event is the ratio of the number of elements in the event to the total number of possible elements. Find the event in the above question. Find the total number of elements. And then use the formula of basic theorem of probability.

Complete step-by-step answer:
Let us say that $ S $ is a sample space of all possible outcomes. And $ n(S) $ is the total number of possible outcomes.
Let us say that $ E $ is an event will possible outcomes of that event. And $ n(E) $ is the total number of possible outcomes of that event.
Then, according to the basic theorem of probability, the probability of occurrence of an element in the said event is given by $ P(E) $ .
Where,
 $ P(E) = \dfrac{{n(E)}}{{n(S)}} $ . . . (1)
Sample space for this question will be the collection of cards from 1 to 30. i.e.
 $ S = \{ 1,2,3....,30\} $
 $ \Rightarrow n(S) = 30 $
(i) a multiple of 4 or 6
For this, the event can be written as
 $ E = \{ 4,6,8,12,16,18,20,24,28,30\} $
 $ \Rightarrow n(E) = 10 $
Therefore, from equation (1), we get
 $ P(E) = \dfrac{{10}}{{30}} = \dfrac{1}{3} $
Thus, the probability of getting a card which is a multiple of 4 or 6 is $ \dfrac{1}{3} $
So, the correct answer is “ $ \dfrac{1}{{3}} $ ”.

(ii) a multiple of 3 and 5
For this, the event can be written as
 $ E = \{ 15,30\} $
 $ \Rightarrow n(E) = 2 $
Therefore, from equation (1), we get
 $ P(E) = \dfrac{2}{{30}} = \dfrac{1}{{15}} $
Thus, the probability of getting a card which is a multiple of 3 and 5 is $ \dfrac{1}{{15}} $
So, the correct answer is “ $ \dfrac{1}{{15}} $ ”.

(iii) a multiple of 3 or 5
For this, the event can be written as
 $ E = \{ 3,5,6,9,10,12,15,18,20,21,24,25,27,30\} $
 $ \Rightarrow n(E) = 14 $
 $ P(E) = \dfrac{{14}}{{30}} = \dfrac{7}{{15}} $
Thus, the probability of getting a card which is a multiple of 3 or 5 is $ \dfrac{7}{{15}} $
So, the correct answer is “ $ \dfrac{7}{{15}} $ ”.

Note: In this question, knowing the basic theorem of probability and knowing how to differentiate between an event and a sample space is important. Once you understand that and can find the number of elements in the event as well as in sample space. Then this question is about just substituting the values in the formula.