Question

Consider a number W = 2³ × 3^α+1 × 5² × 7² × 11^α, α $\in$ N

A set 'S' contains divisor of W. If a divisor is randomly selected from the set S, three events A, B, C are defined as

A : selected divisor is a multiple of 10

B : selected divisor is divisible by 3

C : selected divisor is a multiple of 11

If P(C) = $\frac{3}{4}$ , then answer the following questions.

[Note: P(E) represents probability of event E]

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Linked Question 1

The value of $P (\frac{B}{A}) + P (\frac{C}{A})$ is equal to

$\frac{3}{5}$

$\frac{3}{4}$

$\frac{4}{5}$

$\frac{31}{20}$

$∵ P (C) = \frac{3}{4}$

$\Rightarrow P (C) = \frac{4 \times (α + 2) \times 3 \times 3 \times α}{4 \times (α + 2) \times 3 \times 3 \times (α + 1)} = \frac{3}{4}$

⇒ 4α = 3α + 3 ⇒ α = 3

(i) $P (\frac{B}{A}) + P (\frac{C}{A}) = \frac{P (A \cap B)}{P (A)} + \frac{P (C \cap A)}{P (A)}$

$= \frac{3 \times 4 \times 2 \times 3 \times 4}{3 \times 5 \times 2 \times 3 \times 4} + \frac{3 \times 5 \times 2 \times 3 \times 3}{3 \times 5 \times 2 \times 3 \times 4} = \frac{4}{5} + \frac{3}{4} = \frac{31}{20}$

(ii) $P (\bar{A} \cap B \cap C) = P (B \cap C) - P (A \cap B \cap C)$

$= \frac{4 \times 4 \times 3 \times 3 \times 3}{4 \times 5 \times 3 \times 3 \times 4} - \frac{3 \times 4 \times 2 \times 3 \times 3}{4 \times 5 \times 3 \times 3 \times 4} = \frac{3}{5} - \frac{3}{10} = \frac{3}{10} .$

Linked Question 2

The value of $P (\bar{A} \cap B \cap C)$ is equal to

$\frac{3}{10}$

$\frac{8}{9}$

$\frac{7}{9}$

$\frac{2}{5}$

$∵ P (C) = \frac{3}{4}$

$\Rightarrow P (C) = \frac{4 \times (α + 2) \times 3 \times 3 \times α}{4 \times (α + 2) \times 3 \times 3 \times (α + 1)} = \frac{3}{4}$

⇒ 4α = 3α + 3 ⇒ α = 3

(i) $P (\frac{B}{A}) + P (\frac{C}{A}) = \frac{P (A \cap B)}{P (A)} + \frac{P (C \cap A)}{P (A)}$

(ii) $P (\bar{A} \cap B \cap C) = P (B \cap C) - P (A \cap B \cap C)$