Question

A box contains card numbers from 11 to 123. A card drawn at random from the box. Find the probability that the number on the card is a multiple of 7.

Answer 1

Hint: We know that the ratio of favourable outcomes to total number of outcomes is called probability. It was given that A box contained card numbers from 11 to 123. Now we should find the total numbers from 11 to 123 which are divisible by 7. Now we should know how many numbers are divisible by 7. This represents the total number of favourable outcomes. Now we have to find the total number of outcomes. Now by using this, we can find the probability that the number is divisible by 7.

Complete step-by-step answer:
Before solving the question, we should know the definition of probability. The ratio of favourable outcomes to total number of outcomes is called probability.
From the question, we were given cards marked from 11 to 123. From the numbers 11 to 123, we have to find the numbers divisible by 7.
We know that the least number which is divisible by 7 greater than 11 is equal to 14 and the largest number which is divisible by 7 below 123 is 119. Now we should find the count of numbers from 14 to 119.
We know that if a is the first of A.P, b is the last term of A.P , d is the common difference of A.P , and n is the number of terms then \[b=a+(n-1)d\].
We know that the first term is equal to 14 and the last term is equal is equal to 119 and the common difference is equal to 7.
Now we get
\[\begin{align}
  & \Rightarrow 119=14+(n-1)7 \\
 & \Rightarrow 105=(n-1)7 \\
 & \Rightarrow n-1=15 \\
 & \Rightarrow n=16 \\
\end{align}\]
So, it is clear that the value of n is equal to 16. Hence, the count of numbers from 11 to 123 which are divisible by 7 are equal to 16. Hence, we can say that the total number of outcomes are equal to 16.
Now we should find the total number of outcomes. The given cards are marked as 11 to 123. So, the total number of cards is equal to 113.
We know that the ratio of favourable outcomes to total number of outcomes is called probability. Let us assume the probability that the number is divisible by 7 is P(E).
\[\Rightarrow P(E)=\dfrac{16}{113}\]
So, the probability that the number is divisible by 7 is equal to \[\dfrac{16}{113}\].

Note: Students may also consider that 7 as one of the favourable outcomes unknowingly. If we consider 7 as a favourable outcome, then we get the total number outcomes is equal to 17. We know that the total number of outcomes are equal to 16.
We know that the ratio of favourable outcomes to total number of outcomes is called probability. Let us assume the probability that the number is divisible by 7 is P(E).
\[\Rightarrow P(E)=\dfrac{17}{113}\]
So, the probability that the number is divisible by 7 is equal to \[\dfrac{17}{113}\]. But we know that the probability that the number is divisible by 7 is equal to \[\dfrac{16}{113}\].