Q.Prove that if $E$ and $F$ are independent events, then so are the events $E$ and $F'$.

Concept: Probability Complement Rule — If $E$ and $F$ are independent, then $P(E \cap F) = P(E)P(F)$. We use this to show $P(E \cap F') = P(E)P(F')$.

Step 1: Write $E$ as the disjoint union $(E \cap F) \cup (E \cap F')$.

Thus $P(E) = P(E \cap F) + P(E \cap F')$.

Step 2: Substitute $P(E \cap F) = P(E)P(F)$ (independence) into the equation:

$P(E) = P(E)P(F) + P(E \cap F')$.

Step 3: Rearrange to isolate $P(E \cap F')$:

$P(E \cap F') = P(E) - P(E)P(F) = P(E)[1 - P(F)] = P(E)P(F')$. …

The idea is simple: independence means knowing $F$ happened tells you nothing about $E$. If that’s true, then knowing $F$ didn’t happen should also tell you nothing about $E$. The proof just uses the complement rule and the definition of independence.

1. Start with what we know: $E$ and $F$ are independent. By definition,

2. We want to check whether $E$ and $F'$ are independent. That means we need to verify

3. The event $E$ can be split into two disjoint parts: outcomes where $F$ happens, and outcomes where $F$ doesn’t happen. So

These two pieces are mutually exclusive (they can’t both occur), so their probabilities add:

4. Rearranging gives

5. Now substitute the independence condition $P(E \cap F) = P(E) \cdot P(F)$:

6. But $1 - P(F)$ is exactly $P(F')$ (the complement rule). So

$P(E \cap F') = P(E) \cdot P(F').$ …

Method: Proving complements of independent events are independent

Use this pattern for any "prove independence" result — split an event into disjoint pieces, then substitute the independence condition.

Steps

Step 1: State what independence you must prove.

To show $E$ and $F'$ are independent you must establish the target identity

Step 2: Split the known event into disjoint parts.

Write $E$ as the disjoint union of "$E$ with $F$" and "$E$ without $F$":

$E = (E\cap F)\cup(E\cap F') \;\Longrightarrow\; P(E) = P(E\cap F) + P(E\cap F').$ …

Common Mistakes

Mistake 1: Assuming $P(E\cap F')=P(E)P(F')$ at the start.

Why it's wrong: that equation is the very thing to be proved; assuming it is circular. Correct approach: derive it from the disjoint split of $E$ plus the given $P(E\cap F)=P(E)P(F)$.

Mistake 2: Not using that $E\cap F$ and $E\cap F'$ are disjoint. …