Question

A bag contains 50 coins and each coin is marked from 51 to 100. If one coin is picked at random then the probability that the number on the coin is not a prime number is:
(a)$\dfrac{1}{5}$
(b)$\dfrac{3}{5}$
(c)$\dfrac{2}{5}$
(d)$\dfrac{4}{5}$

Answer 1

It is given that a bag contains 50 coins and each coin is marked from 51 to 100. We are asked to find the probability of drawing a coin from the bag in such a way that the drawn coin is not a prime number. First of all, find the number of prime numbers from 51 to 100. Now, we want that the drawn number is not a prime number so subtract the number of prime numbers from the total numbers (i.e. 50). The subtraction is a favorable outcome. Now, we know that to find the probability of a certain outcome we are going to use the following formula $\dfrac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$. We have already shown how favorable outcomes will come and the total outcomes are equal to 50. Substitute these values of outcomes and get the required probability.

Complete step-by-step solution:
There are 50 coins in a bag. The numbering on the coins is from 51 to 100.
We are asked to find the probability of drawing a coin which is not a prime number.
We know that the probability of an outcome is equal to:
$\dfrac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$
Total outcomes in the above formula are equal to the total number of coins which are 50.
And the favorable outcomes are the number, not prime numbers from 51 to 100.
To get not prime numbers, we have to first find the number of prime numbers and then subtract these numbers of prime numbers from the total numbers.
Prime numbers from 51 to 100 are as follows:
53, 59, 61, 67, 71, 73, 79, 83, 87, 97
As you can see that we are getting 10 prime numbers from 51 to 100 so to find the not prime numbers we are going to subtract 10 from the total numbers (i.e. 50).
$\begin{aligned} & 50-10 \\\ & =40 \\\ \end{aligned}$
Hence, we get not prime numbers as 40. And not prime numbers are our favorable outcomes in the probability formula.
The probability of drawing a number which is not a prime number is equal to:
$\dfrac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$
Substituting favorable outcomes as 40 and total outcomes as 50 in the above expression we get,
$\begin{aligned} & \dfrac{40}{50} \\\ & =\dfrac{4}{5} \\\ \end{aligned}$
Hence, we got the probability of drawing a number from the bag which is not a prime number is $\dfrac{4}{5}$.
Hence, the correct option is (d).

Note: The alternative way to solve the above problem is as follows:
First, find the probability of drawing a prime number and then subtract this probability from 1to get the probability of drawing not a prime number from the bag.
The number of prime numbers we have already shown above as 10.
Then the probability of drawing a prime number is equal to:
$\dfrac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$
Now, in this method, favorable outcomes become 10 and total is 50 so substituting these values in the above we get,
$\begin{aligned} & \dfrac{10}{50} \\\ & =\dfrac{1}{5} \\\ \end{aligned}$
There is a property of the probability that:
$P\left( E \right)+P\left( notE \right)=1$
Here, E is the event of drawing a prime number so probability of drawing not a prime number is the subtraction of probability of drawing a prime number from 1.
$\begin{aligned} & 1-\dfrac{1}{5} \\\ & =\dfrac{4}{5} \\\ \end{aligned}$