Question:
An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two
non-empty events of that experiment. If $A$ consists of $4$ outcomes,
find the number that $B$ must have in order for $A$ and $B$ to be
independent.
I really don't understand what the question is asking. For two events to be independent, the probability of one of the events occurring must be independent of the other event occurring. So shouldn't the answer simply be $6$?
Best Answer
It sounds like you're confusing independent with mutually exclusive. Events $A$ and $B$ are independent if $P(A\cap B)=P(A)P(B)$. Events $A$ and $B$ are mutually exclusive if $A \cap B = \varnothing$, i.e., whenever one of $A$ or $B$ occurs, the other does not.
This problem asks you to determine what the number of outcomes in $B$ must be in order to have $P(A \cap B)=P(A)P(B)$. Since the experiment has $10$ equally likely outcomes,
\begin{align*} P(A \cap B)= \frac{|A \cap B|}{10},\ P(A)=\frac{|A|}{10}=\frac{4}{10}, \ P(B)=\frac{|B|}{10}. \end{align*}
The equation $P(A\cap B)=P(A)P(B)$ then becomes $\frac{|A\cap B|}{10}=\frac{4}{10}\cdot \frac{|B|}{10}$, or $5\cdot |A\cap B|=2 \cdot |B|$. What are the possible values of $|B|$ that could satisfy this equation?