Question

There are $40$ tickets numbered from $1 - 40$ , one ticket is drawn at random. Find the probability of getting $\left( a \right)$ divisible of $4$ and $\left( b \right)$ prime numbers.

Answer 1

Formula for probability is the ratio of favourable outcomes by the total number of outcomes. So start with finding the number of ticket numbers that are divisible by four and divide it by $40$ to find the correct answer. Then again for part $\left( b \right)$ find the number of primes till the ticket number $40$. Divide the number of primes by the total number of tickets to get the required probability.

Complete step-by-step answer:
Here in this problem, we are given with forty tickets numbers from one to forty and from this group of tickets, one ticket is drawn out random. From this information, we need to find the probability of getting a ticket with a number divisible by $4$ on it and the probability of getting a prime number.
Before starting with the solution let us first understand the concept of probability. 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.
$\Rightarrow$ Probability $= \dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of outcomes}}}}$
So, for finding the required probability we need to first find the number of favourable outcomes and then divide it by the total number of outcomes.
$\Rightarrow$ Numbers divisible by $4$ and less than $40$ $= 4,8,12,16,20,24,28,32,36{\text{ and }}40$
Therefore, the favourable outcomes for the case $\left( a \right)$ will be $4,8,12,16,20,24,28,32,36{\text{ and }}40$
$\Rightarrow$ The number of favourable outcomes $= 10$
Now we can use the formula for finding the required probability
$\Rightarrow$ Probability of getting a number divisible by $4$ $= \dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of outcomes}}}} = \dfrac{{10}}{{40}} = \dfrac{1}{4}$
For part $\left( b \right)$ we need to find the favourable outcomes first:
$\Rightarrow$ The prime numbers in $1 - 40$ticket numbers are $2,3,5,7,11,13,17,19,23,29,31{\text{ and }}37$
Therefore, the number of favourable outcomes for the required probability is $12$
Now we can use the formula for finding the probability
$\Rightarrow$ Probability of getting a prime number $= \dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of outcomes}}}} = \dfrac{{12}}{{40}}$
This can be further simplified as:
$\Rightarrow$ Probability of getting a prime number $= \dfrac{{12}}{{40}} = \dfrac{6}{{20}} = \dfrac{3}{{10}}$
Hence, we get both the required probability as $\dfrac{1}{4}$ and $\dfrac{3}{{10}}$.

Note: In questions like this, always write the formula for probability and try to go step by step by finding favourable outcomes and the total number of outcomes. The prime numbers are the numbers which are only divisible by $1$ and by themselves. The prime numbers start with $2$ . We can find the prime numbers by the trial method. Remember that the answer for any probability will always be less than one.