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# Let E and F be two independent events. The probability that both E and F happen is $\dfrac{1}{12}$ and the probability that neither E nor F happens is $\dfrac{1}{2}$, then a value of $\dfrac{P\left( E \right)}{P\left( F \right)}$ is(a) $\dfrac{5}{12}$(b) $\dfrac{1}{3}$(c) $\dfrac{3}{2}$(d) $\dfrac{4}{3}$  Verified
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Hint: Use the two given probabilities to make two equations. Then, using the formula $P\left( E\cup F \right)=P\left( E \right)+P\left( F \right)-P\left( E\cap F \right)$ and $P\left( E\cap F \right)=P\left( E \right)\cdot P\left( F \right)$, make two equations and solve them to find the values of $P\left( E \right)$ and $P\left( F \right)$.

We know the following facts:
1. The probability that two events A and B happen together is given as $P\left( A\cap B \right)$
2. The probability that at least one of the two events A and B happens is given as $P\left( A\cup B \right)$
3. The probability that an event E does not happen is given as $1-P\left( E \right)$, if $P\left( E \right)$ is the probability that the event A happens.
Applying the above facts to the statements given in the question:
Probability that E and F happen together is $\dfrac{1}{12}$, which can be written as $P\left( E\cap F \right)=\dfrac{1}{12}$
The second statement, probability that neither E nor F happen can be understood as the negation of the event that at least one of them happens.
The probability that at least one of E or F happens is given as $P\left( E\cup F \right)$.
Hence, the probability of neither E nor F happens is given as $1-P\left( E\cup F \right)=\dfrac{1}{2}$. Upon rearranging,
\begin{align} & \Rightarrow P\left( E\cup F \right)=1-\dfrac{1}{2} \\ & \Rightarrow P\left( E\cup F \right)=\dfrac{1}{2} \\ \end{align}
Thus, we have two results $P\left( E\cap F \right)=\dfrac{1}{12}$ and $P\left( E\cup F \right)=\dfrac{1}{2}$.
We know that $P\left( E\cup F \right)=P\left( E \right)+P\left( F \right)-P\left( E\cap F \right)$.
Substituting the value of $P\left( E\cup F \right)$ and $P\left( E\cap F \right)$ in the above formula, we get
\begin{align} & \dfrac{1}{2}=P\left( E \right)+P\left( F \right)-\dfrac{1}{12} \\ & \Rightarrow P\left( E \right)+P\left( F \right)=\dfrac{1}{2}+\dfrac{1}{12} \\ & \Rightarrow P\left( E \right)+P\left( F \right)=\dfrac{7}{12}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots \left( 1 \right) \\ \end{align}
Also, since the events E and F are independent, $P\left( E\cap F \right)=P\left( E \right)\cdot P\left( F \right)$
Thus, $P\left( E \right)\cdot P\left( F \right)=\dfrac{1}{12}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots \left( 2 \right)$
To solve the equations (1) and (2) to find $P\left( E \right)$ and $P\left( F \right)$, we can use the relation $a-b=\sqrt{{{\left( a+b \right)}^{2}}-4ab}$
In this equation, $a=P\left( E \right)$ and $b=P\left( F \right)$
$\mathop{\left( P\left( E \right)-P\left( F \right) \right)}^{2}={{\left( P\left( E \right)+P\left( F \right) \right)}^{2}}-4P\left( E \right)\cdot P\left( F \right)$
Substituting values from equations (1) and (2),
\begin{align} & \Rightarrow \mathop{\left( P\left( E \right)-P\left( F \right) \right)}^{2}={{\left( \dfrac{7}{12} \right)}^{2}}-4\left( \dfrac{1}{12} \right) \\ & \Rightarrow \mathop{\left( P\left( E \right)-P\left( F \right) \right)}^{2}=\left( \dfrac{49}{144} \right)-\left( \dfrac{1}{3} \right) \\ & \Rightarrow \mathop{\left( P\left( E \right)-P\left( F \right) \right)}^{2}=\dfrac{1}{144} \\ & \Rightarrow P\left( E \right)-P\left( F \right)=\dfrac{1}{12}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots \left( 3 \right) \\ \end{align}
$2\cdot P\left( E \right)=\dfrac{8}{12}$
$\Rightarrow P\left( E \right)=\dfrac{4}{12}=\dfrac{1}{3}$
$2\cdot P\left( F \right)=\dfrac{6}{12}$
$\Rightarrow P\left( F \right)=\dfrac{3}{12}=\dfrac{1}{4}$
Thus, the required value, $\dfrac{P\left( E \right)}{P\left( F \right)}=\dfrac{\dfrac{1}{3}}{\dfrac{1}{4}}=\dfrac{4}{3}$
Note: The formula used here, $P\left( E\cap F \right)=P\left( E \right)\cdot P\left( F \right)$ is only valid if the two events E and F are independent of each other (given in the question). Otherwise this formula is not applicable, and then using this formula would result in an incorrect answer.