Sets and probability The number of total outcomes of an experiment are $25$. If $A$ and $B$ are two non-empty independent events of the experiment such that outcomes in favour of event $A$ are $15$, then the minimum number of outcomes in favour of event $B$ can be?

Let's suppose that $B$ has $b \gt 0$ equally probable favourable outcomes, while $A$ has $15$.

Then $P(A \cap B) = P(A)P(B) = \frac{15}{25} \times \frac{b}{25} = \frac{3b}{125}$ given $A$ and $B$ are independent. But since there are only $25$ possibilities for $A \cap B$, $3b$ must be a multiple of $5$, so $B$ must have $5$, $10$, $15$, $20$ or $25$ favourable outcomes.

The smallest of these is $5$, as Did seems to have commented while I was typing.