First slide

Theorems of probability

Question

$A positive integer n not exceeding 100 is chosen at random in such a way that, p is the$
$probability of choosing n \leq 50 and 3 p is the probability of choosing n > 50 . If λ is$
$probability that a perfect square is chosen, then the value of [100 λ] ([.] \to GIF)$

Moderate

Solution

$Since the probability is p for choosing n \leq 50 and 3 p for choosing n \geq 50, then$
$\begin{array}{l} 50 p + 50 (3 p) = 1 \Rightarrow p = 0.05 \\ \Rightarrow 50 p + 150 p = 1 \\ \Rightarrow 200 p = 1 \\ \Rightarrow p = \frac{1}{200} = 0.05 \\ ∴ Required probability is 7 p + 3 (3 p) = 16 p = 0.08 \end{array}$
$∴ 100 λ = 8$

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