Question

Let $E$ be an event which is neither a certainty nor an impossibility. If probability is such that $P\left(E\right)=1+\lambda +{\lambda }^2$ and $P\left(E^{\prime }\right)=(1+\lambda {)}^2$ in terms of an unknown $\lambda$, then $P\left(E\right)$ is equal to

• A. $1$
• B. $3/4$
• C. $1/4$
• D. none of these

Answer 1

The correct option is B $3/4$

Let $E$ be an event which is neither certainty nor an impossibility.

Given, $P\left(E\right)=1+\lambda +{\lambda }^2$

$\Rightarrow P\left(E^{\prime }\right)={(1+\lambda )}^2$

$\Rightarrow P\left(E\right)+P\left(E^{\prime }\right)=0$

$\Rightarrow 1+\lambda +{\lambda }^2+{(1+\lambda )}^2=0$

$1+\lambda +{\lambda }^2+1+2\lambda +{\lambda }^2=0$

$2+3\lambda +{2\lambda }^2=0$

$\Rightarrow 3\lambda =-(2+{2\lambda }^2)$

$\Rightarrow \frac{-3\lambda }{2}=1+{\lambda }^2$

Neglect negative sigh, put $\lambda =1$

$\Rightarrow \lambda =\frac{3}{4}.$

Hence, the answer is $\frac{3}{4}.$