Question

An experiment has $10$ equally likely outcomes. let $A\&B$be two non-empty events of the experiment. if $A$ consists of $4$ outcomes that $B$ must have so that $A\&B$ are independent, is equal to

• A. $2,4or8$
• B. $3,6or9$
• C. $4or8$
• D. $5or10$

Answer 1

The correct option is D

$5or10$

Finding the event:

Given, An experiment has $10$ equally likely outcomes. let $A\&B$be two non-empty events of the experiment. if $A$ consists of $4$ outcomes,

The number of outcomes that $B$ must have

so that $A\&B$ are independent, is equal to

$$\begin{gathered} \Rightarrow P\left(A\cap B\right)=P\left(A\right).P\left(B\right) \\ \Rightarrow \frac{\mathrm{n}(\mathrm{A}\cap \mathrm{B})}{10}=\frac{\mathrm{n}\left(\mathrm{A}\right)}{10}.\frac{\mathrm{n}\left(\mathrm{B}\right)}{10} \\ \Rightarrow \frac{n\left(A\cap B\right)}{10}=\frac{4}{10}.\frac{n\left(B\right)}{10} \\ \Rightarrow \frac{n\left(A\cap B\right)}{10}=\frac{2}{5}.\frac{n\left(B\right)}{10} \\ \Rightarrow 5n(\mathrm{A}\cap \mathrm{B})=2.n\left(\mathrm{B}\right) \\ \Rightarrow n\left(B\right)=5or10 \end{gathered}$$

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 5 or 10

Hence option $\left(D\right)$ is correct.