Question

An experiment has $24$ equally likely outcome . let A and B two non-empty events of the experiments . If B consists of $9$ outcome the number of outcomes that A must have so that A and B are independent events is

• A. $8$ or $16$ or $24$
• B. $2$ or $4$ or $8$
• C. $3$ or $6$ or $9$
• D. $4$ or $8$ or $12$

Answer 1

The correct option is A $8$ or $16$ or $24$

Since A and B are independent events,
$P(A\cap B)=P\left(A\right)\times P\left(B\right)$

$\frac{n(A\cap B)}{n\left(S\right)}=\frac{n\left(A\right)}{n\left(S\right)}\times \frac{n\left(B\right)}{n\left(S\right)}$

$\therefore$ $n\left(S\right).n(A\cap B)=n\left(A\right)n\left(B\right)$

24$.n(A\cap B)=n\left(A\right).9$

8 $n(A\cap B)=n\left(A\right).3$

Since $n\left(A\right),n(A\cap B)$ are $+$ ve integers and $n(A\cap B)\le n\left(A\right)$ as well as 9 $n\left(A\right)$ can be $8$ or $16$ or $24$