Question

An experiment has $10$ equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of $4$ outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is

• A. $2,4$ or $8$
• B. $3,6$ or $9$
• C. $4$ or $8$
• D. $5$ or $10$

Answer 1

The correct option is D $5$ or $10$

Since the Set $A$ has $4$ out of $10$ outcomes, we have $P\left(A\right)=\frac{4}{10}$

Similarly, let set $B$ has $k$ outcomes, then we have $P\left(B\right)=\frac{k}{10}$
Now, let $A$ and $B$ have $m$ outcomes in common i.e., $\left|A\cap B\right|=m$, then $m\le \left|A\right|=4$.

Hence,

$P\left(A\cap B\right)=\frac{m}{10}$.
Now, $P\left(A|B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{\frac{m}{10}}{\frac{k}{10}}=\frac{m}{k}$
But, $P\left(A\cap B\right)=P\left(A\right)\times P\left(B\right)$ as $A$ and $B$ are independent.

$\therefore P\left(A|B\right)=P\left(A\right)=\frac{4}{10}$
Hence, $\frac{m}{k}=\frac{4}{10}=\frac{2}{5}$

$2k=5m$
The only integral answers for $m$ are $0,2,$ and $4$.

If $m=0\Rightarrow k=0$, contrary to the given fact that $B$ is a non-empty event.
Therefore, $m$ must be equal to either to either $2$ or $4$, which gives us values of $k$ as $5$ or $10$.