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 $10$

Let the outcomes of set B be x.
Now A and B are independent sets.
Hence
$P(A\cap B)$
$=P\left(A\right).P\left(B\right)$
$=\frac{4}{10}.\frac{x}{10}$
$=\frac{4x}{100}$
$=\frac{\frac{2x}{5}}{10}$
Hence
$\frac{2x}{5}$ is an integer and
Thus x is a multiple of 5.