Question

In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays. Find the probability that, on a ball played, he does not hit a boundary.

Answer 1

Hint: For the above question of probability, we will use the property that, the probability of all mutually exclusive and exhaustive events is equal to 1. Here, the probability of hitting the boundary and the probability of not hitting the boundary are two mutually exclusive and exhaustive events, so, their sum must be equal to 1. From the question, we will get the favorable outcomes and total outcomes, and then the ratio of these will give us a probability of hitting a boundary.

Complete step-by-step answer:
We have been given that, in a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays and we have to find the probability of not hitting boundary.
So, the probability of hitting boundary\[\Rightarrow \dfrac{\text{Number of boundaries hit}}{\text{Total number of balls}}=\dfrac{6}{30}=\dfrac{1}{5}\].

We know that the sum of two mutually exclusive and exhaustive events is equal to 1.
We have the probability of hitting boundary and the probability of not hitting boundary are two mutually exclusive and exhaustive events, so, their sum must be equal to 1.
\[\begin{align}
  & \Rightarrow P\left( \text{hitting boundary} \right)+P\left( \text{not hitting boundary} \right)=1 \\
 & \Rightarrow \dfrac{1}{5}+P\left( \text{not hitting boundary} \right)=1 \\
 & \Rightarrow P\left( \text{not hitting boundary} \right)=1-\dfrac{1}{5} \\
 & \Rightarrow P\left( \text{not hitting boundary} \right)=\dfrac{5-1}{5} \\
\end{align}\]
\[\Rightarrow P\left( \text{not hitting boundary} \right)=\dfrac{4}{5}\]

Therefore, the probability of not hitting a boundary is equal to \[\dfrac{4}{5}.\]

Note: We can also find the probability that, on a ball a player doesn't hit a boundary by calculating the number of ball on which a player doesn't hit a boundary out of 30 balls and the probability is equal to the ratio of number of balls a batsman doesn't hit a boundary to total number of balls. So, we have a favorable outcome as $30-6=24$ and total outcome as 30. Then, the probability of not hitting a boundary \[\Rightarrow \dfrac{24}{30}=\dfrac{4}{5}\]