A random experiment is an experiment whose outcome cannot be predicted with certainty before it is performed. For example, when we toss a coin, we cannot predict whether it will land on heads or tails. The sample space is the set of all possible outcomes. For a coin toss, the sample space is H and T. For rolling a six-sided die, the sample space contains the numbers 1 through 6.
An event is a subset of the sample space. There are different types of events. A simple event contains only one outcome, like getting exactly 3 when rolling a die. A compound event contains multiple outcomes, such as getting an even number which includes 2, 4, and 6. An impossible event is the empty set that can never occur, while a certain event is the entire sample space that always occurs.
Mutually exclusive events are events that cannot occur at the same time. Their intersection is the empty set. For example, when tossing a coin, getting heads and getting tails are mutually exclusive because a coin cannot land on both sides simultaneously. In a Venn diagram, mutually exclusive events are represented as non-overlapping circles within the sample space.
Exhaustive events are a set of events whose union covers the entire sample space. This means that at least one of these events must occur in any trial of the experiment. For example, when rolling a die, the events getting an odd number and getting an even number are exhaustive because together they include all possible outcomes from 1 to 6.
In summary, we have covered the fundamental concepts of probability theory. A random experiment has uncertain outcomes, with all possibilities forming the sample space. Events are subsets of this space, ranging from simple single-outcome events to compound multi-outcome events. We also learned about mutually exclusive events that cannot occur together, and exhaustive events that cover all possibilities. These concepts provide the essential foundation for understanding probability and statistical analysis.