First slide

Introduction to probability

Question

Let E be an event which is neither a certainty nor an impossibility. If probability is such that $P (E) = 1 + λ + λ^{2}$ and $P (E') = (1 + λ)^{2}$ in terms of an unknown $λ$ . Then P(E) is equal to

Easy

A

1
B

$\frac{3}{4}$
C

$\frac{1}{4}$
D

none of these

Solution

$P (E) + P (E') = 1$
$∴ 1 + λ + λ^{2} + (1 + λ)^{2} = 1$
$\Rightarrow 2 λ^{2} + 3 λ + 1 = 0$
$\Rightarrow (2 λ + 1) (λ + 1) = 0$
$\Rightarrow λ = - 1, - \frac{1}{2}$
For $λ = - 1, P (E) = 1 + (- 1) + (- 1)^{2} = 1$ (not possible)
For $λ = - \frac{1}{2}, P (E) = 1 + (- \frac{1}{2}) + {(- \frac{1}{2})}^{2} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}$

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