Question

Tickets numbered from 1 to 20 are mixed up and a ticket is drawn at random. The probability that the ticket drawn has a number which is a multiple of 3 or 7 is $\dfrac{2}{{\text{x}}}$ . What is the value of x.

Answer 1

Hint: In this question, first we will find the multiples of 3 and 7 which are lying between 1 to 20. After this, we will use the formula for finding the probability which is given as the ratio of total favourable event and total events. The favourable event is obtained by adding the multiples of 3 and 7 between 1 to 20.

Complete step-by-step answer:
In the question, it is given that tickets numbered from 1 to 20 are mixed up and drawn at random.
Tickets to be drawn are multiples of 3 and 7.
To find the probability, we will the formula for probability which is given as:
Probability =${\text{ }}\dfrac{{{\text{Total numbers of favourable events}}}}{{{\text{Total numbers of possible events}}}}$ .
In this question, the total number of favourable events is equal to the total number multiples of 3 and 7 between 1 to 20.
So, we will now find the multiples of 3 and 7 between 1 to 20.
Multiples of 3 between 1 to 20 are: 3, 6, 9, 12, 15, 18.
Total numbers of multiples of 3 = 6
Multiples of 7 between 1 to 20 are: 7, 14.
Total numbers of multiples of 7=2
So, total numbers of favourable events = sum of multiples of 3 and 7= 6+2=8.
And total numbers of possible events = 20
So we can write:
Probability =${\text{ }}\dfrac{{{\text{Total numbers of favourable events}}}}{{{\text{Total numbers of possible events}}}} = \dfrac{8}{{20}}$ .
Now, in the question probability is given. So, we can write:
$
  \dfrac{2}{{\text{x}}} = \dfrac{8}{{20}} \\
  {\text{We can write above equation as:}} \\
  \dfrac{{\text{x}}}{2} = \dfrac{{20}}{8} \\
   \Rightarrow {\text{x = }}\dfrac{{20}}{8} \times 2 = 5 \\
   \Rightarrow {\text{x = 5}}{\text{.}} \\
$
Therefore, the value of x is 5.

Note: In this type of question, you should be careful with words like ‘OR’ and ‘AND’. Here in this question ‘OR’ is given. So, we will add the multiples of 3 and 7 to find the total numbers of favourable events. In case if ‘AND’ is given then we have to find common multiples of 3 and 7 to find total numbers of favourable events.