First slide

Theorems of probability

Question

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

Moderate

A

1
B

$\frac{3}{4}$
C

$\frac{1}{4}$
D

None of these

Solution

$P (E) + P (E^{'}) = 1 = 1 + λ + λ^{2} + (1 + λ)^{2}$
$\begin{array}{l} or 2 λ^{2} + 3 λ + 1 = 0 \\ or (2 λ + 1) (λ + 1) = 0 \\ or λ = - 1, - \frac{1}{2} \end{array}$
Then,
$\begin{array}{l} P (E) = 1 + (- 1) + (- 1)^{2} = 1 (not possible) \\ = 1 + (- \frac{1}{2}) + {(- \frac{1}{2})}^{2} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \end{array}$

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