Question

If $A$ is an event in a random experiment such that $P\left( A \right):P\left( {\bar{A}} \right)=5:11$, then find $P\left( A \right)$ and $P\left( {\bar{A}} \right)$.

Answer 1

In this question we have been given with the ratio of the probabilities of an event $A$ and its complement given to us as $P\left( A \right):P\left( {\bar{A}} \right)=5:11$ and from this data we have to find the value of $P\left( A \right)$ and $P\left( {\bar{A}} \right)$. We will do this by using the property of probability that the probability of the compliment is $1$ minus the probability of the event occurring which can be written as $P\left( {\bar{A}} \right)=1-P\left( A \right)$. We will then use the ratio given to us and substitute the value of $P\left( {\bar{A}} \right)$ in terms of $P\left( A \right)$ and solve for the value of $P\left( A \right)$. We will then substitute the value of $P\left( A \right)$ in $P\left( {\bar{A}} \right)=1-P\left( A \right)$ to get the value of $P\left( {\bar{A}} \right)$ and write the required solution.

Complete step by step solution:
We have the ratio given to us as:
$\Rightarrow P\left( A \right):P\left( {\bar{A}} \right)=5:11$
We can write them in the fraction format as:
$\Rightarrow \dfrac{P\left( A \right)}{P\left( {\bar{A}} \right)}=\dfrac{5}{11}$
Now we know the property of probability that
$\Rightarrow P\left( {\bar{A}} \right)=1-P\left( A \right)\to \left( 1 \right)$
Therefore, on substituting the value, we get:
$\Rightarrow \dfrac{P\left( A \right)}{1-P\left( A \right)}=\dfrac{5}{11}$
On cross multiplying the terms, we get:
$\Rightarrow 11\times P\left( A \right)=5\times \left( 1-P\left( A \right) \right)$
On simplifying the brackets, we get:
$\Rightarrow 11P\left( A \right)=5-5P\left( A \right)$
On transferring the term $5P\left( A \right)$ from the right-hand side to the left-hand side, we get:
$\Rightarrow 11P\left( A \right)+5P\left( A \right)=5$
On adding the terms, we get:
$\Rightarrow 16P\left( A \right)=5$
On transferring the term $16$ from the left-hand side to the right-hand side, we get:
$\Rightarrow P\left( A \right)=\dfrac{5}{16}$, which is the required probability for the event.
Now on substituting $P\left( A \right)=\dfrac{5}{16}$ in equation $\left( 1 \right)$, we get:
$\Rightarrow P\left( {\bar{A}} \right)=1-\dfrac{5}{16}$
On taking the lowest common multiple, we get:
$\Rightarrow P\left( {\bar{A}} \right)=\dfrac{16-5}{16}$
On simplifying, we get:
$\Rightarrow P\left( {\bar{A}} \right)=\dfrac{11}{16}$, which is the required probability for the compliment event.

Note: It is to be remembered that the complement probability is also called as the probability of the event not happening. If $P\left( A \right)$ represents the probability of an event occurring then $P\left( {\bar{A}} \right)$ is the probability of an event not occurring. It is to be remembered that probability of an event cannot be lesser than $0$ or greater than $1$.