Set Notation is the formal method of grouping mathematical entities that share common features. In probability, the Universal Set (denoted by , , or ) represents the entire sample space containing all possible outcomes.
An Event is a subset of the sample space, usually denoted by capital letters like or . Each individual outcome within that event is called an element.
The Complement of an event , written as , represents all outcomes in the sample space that are NOT in . This leads to the fundamental identity: .
The Intersection () represents the event where BOTH and occur simultaneously. On a Venn diagram, this is the overlapping region between two circles.
The Union () represents the event where occurs, OR 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 (), meaning . In this specific case, the addition rule simplifies to .
Conditional Probability measures the probability of an event occurring, given that event has already occurred. This is denoted as , read as 'the probability of given '.
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 .
The mathematical definition is: where . This shows that we are looking for the portion of that also belongs to .
| Concept | Independent Events | Mutually Exclusive Events |
|---|---|---|
| Definition | One event occurring does not change the probability of the other. | The events cannot happen at the same time. |
| Intersection | ||
| Conditional | $P(A | B) = P(A)$ |
| Visual | Overlapping circles (usually). | Non-overlapping (separate) circles. |
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: . 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.