Conditional Probability

13.2 Conditional Probability

So far we have found probabilities of events using the entire sample space. But real life often gives us partial information: if we know one event has already occurred, does that change the chance of another? This is the central question of conditional probability.

Motivating Example: Tossing Three Fair Coins

Consider tossing three fair coins. The sample space is:

Each outcome is equally likely, with probability $\frac{1}{8}$.

Define two events:

• E: "at least two heads appear"
• F: "first coin shows tail"

Then:

So $P(E) = \frac{4}{8} = \frac{1}{2}$ and $P(F) = \frac{4}{8} = \frac{1}{2}$. The intersection $E \cap F$ contains only THH, so:

Now suppose we are told the first coin shows tail — event F has occurred. The sample space is no longer S: since we know F happened, we restrict attention to the outcomes in F. Within F, only one outcome (THH) is favourable to E, so:

This is the conditional probability of E given F, denoted $P(E|F)$.

Formal Definition of Conditional Probability

From the example:

Dividing numerator and denominator by $n(S)$:

This is valid only when $P(F) \neq 0$.

Properties of Conditional Probability

Property 1: Probability of the Sample Space

Proof: Since $S \cap F = F$,

Property 2: Probability of the Empty Event

Proof: Since $\phi \cap F = \phi$ and $P(\phi) = 0$,

Property 3: Conditional Probability of a Union

For any two events A and B:

Proof:

Using $(A \cup B) \cap F = (A \cap F) \cup (B \cap F)$ and the addition theorem, with $(A \cap F) \cap (B \cap F) = A \cap B \cap F$:

Dividing through by $P(F)$ and writing each term as a conditional probability: