Set Notation & Conditional Probability

Set Notation is the formal method of grouping mathematical entities that share common features. In probability, the Universal Set (denoted by $\xi$, $S$, or $U$) represents the entire sample space containing all possible outcomes.

An Event is a subset of the sample space, usually denoted by capital letters like $A$ or $B$. Each individual outcome within that event is called an element.

The Complement of an event $A$, written as $A'$, represents all outcomes in the sample space that are NOT in $A$. This leads to the fundamental identity: $P(A') = 1 - P(A)$.

The Intersection ($A \cap B$) represents the event where BOTH $A$ and $B$ occur simultaneously. On a Venn diagram, this is the overlapping region between two circles.

The Union ($A \cup B$) represents the event where $A$ occurs, OR $B$ occurs, OR both occur. It encompasses the entire area covered by both circles in a Venn diagram.

For Mutually Exclusive events, the intersection is the Empty Set ($\emptyset$), meaning $P(A \cap B) = 0$. In this specific case, the addition rule simplifies to $P(A \cup B) = P(A) + P(B)$.

Conditional Probability measures the probability of an event $A$ occurring, given that event $B$ has already occurred. This is denoted as $P(A|B)$, read as 'the probability of $A$ given $B$'.

The core effect of the 'given' condition is to reduce the sample space. Instead of looking at the entire universal set, we only consider the outcomes that fall within event $B$.

The mathematical definition is: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ where $P(B) > 0$. This shows that we are looking for the portion of $B$ that also belongs to $A$.

Identify the Denominator: In conditional probability questions, always identify the 'given' condition first. This value becomes your new denominator, replacing the total sample space.

The Addition Formula: Always remember the general addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Students often forget to subtract the intersection, leading to double-counting.

Sanity Checks: Probabilities must always be between 0 and 1. If a conditional probability calculation results in a value greater than 1, you likely used the wrong denominator or forgot to divide by the 'given' event's probability.