Question

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\dfrac{11}{25}$​ and the probability of none of them occurring is $\dfrac{2}{25}$​. If $P(T)$ denotes the probability of occurrence of the event $T$ then,
A. $P(E)=\dfrac{4}{5},P(F)=\dfrac{3}{5}$
B. $P(E)=\dfrac{1}{5},P(F)=\dfrac{2}{5}$
C. $P(E)=\dfrac{2}{5},P(F)=\dfrac{1}{5}$
D. $P(E)=\dfrac{3}{5},P(F)=\dfrac{4}{5}$

Answer 1

Hint: We are given that $P(E,{{F}^{'}})$or $P(F,{{E}^{'}})=\dfrac{11}{25}$. Consider Let $P(E)=x$ and $P(F)=y$, then the probability that none of them occurring will be $P({{E}^{'}},{{F}^{'}})=\dfrac{2}{25}$. Solve it, you will get the answer.

Complete step-by-step answer:

It is given in question that $P(E,{{F}^{'}})$or $P(F,{{E}^{'}})=\dfrac{11}{25}$.
So $P(E,{{F}^{'}})+P(F,{{E}^{'}})=\dfrac{11}{25}$.
Let $P(E)=x$ and $P(F)=y$.
So, we can say that $x(1-y)+y(1-x)=\dfrac{11}{25}$ ……… (1)
$P({{E}^{'}},{{F}^{'}})=\dfrac{2}{25}$
So, we can also say that $(1-x)(1-y)=\dfrac{2}{25}$ ……. (2)
From (2) we get $y=1-\dfrac{2}{25(1-x)}$.
Substituting $y$ in (1) we get,
\[\begin{align}
  & x(1-1+\dfrac{2}{25(1-x)})+\left( 1-\dfrac{2}{25(1-x)} \right)(1-x)=\dfrac{11}{25} \\
 & \dfrac{2x}{25(1-x)}+\left( (1-x)-\dfrac{2}{25} \right)=\dfrac{11}{25} \\
 & \dfrac{2x}{25(1-x)}+(1-x)=\dfrac{13}{25} \\
 & 2x+25{{(1-x)}^{2}}=13(1-x) \\
 & 25(1-2x+{{x}^{2}})=13-15x \\
 & 25{{x}^{2}}-35x+12=0 \\
\end{align}\]
Now solving the quadratic equation we get,
\[25{{x}^{2}}-35x+12=0\]
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
$x=\dfrac{35\pm \sqrt{{{35}^{2}}-4(25)(12)}}{2(25)}=\dfrac{35\pm \sqrt{25}}{2(25)}=\dfrac{35\pm 5}{2(25)}$
So we get, $x=\dfrac{3}{5},\dfrac{4}{5}$. Now substituting these two values of $x$ in the above equation, $y=1-\dfrac{2}{25(1-x)}$, we get, $y=1-\dfrac{2}{25(1-\dfrac{4}{5})}=\dfrac{3}{5}$ and $y=1-\dfrac{2}{25(1-\dfrac{3}{5})}=\dfrac{4}{5}$.
So we get, \[y=\dfrac{3}{5},\dfrac{4}{5}\].
Therefore, $x=\dfrac{4}{5},\dfrac{3}{5}$ and \[y=\dfrac{3}{5},\dfrac{4}{5}\].
Hence, the cases possible are : $P(E)=\dfrac{4}{5},P(F)=\dfrac{3}{5}$ and $P(E)=\dfrac{3}{5},P(F)=\dfrac{4}{5}$.
So the correct answers are options (A) and (D).

Note: Read the question carefully. Your concept regarding independent events should be clear. The occurrence of event E will not affect the occurrence of event F. An example of the same could be tossing a coin and rolling a die at the same time.