Basic Concepts

An event is any subset of the sample space and represents a set of outcomes that share some common feature. Events can be simple (containing one outcome) or compound (containing multiple outcomes).

For example, in a single die roll, the event “roll an even number” is \( E = \{2, 4, 6\} \).

The probability of an event \( A \), written \( P(A) \) (

$P(A)$

) or $\text{Pr}[A]$ (

$\text{Pr}[A]$

), is the sum of the probabilities of the outcomes in \( A \).

If all outcomes in the sample space are equally likely, then \( P(A) = \frac{\text{number of outcomes in } A}{\text{total number of outcomes in } \Omega} \).

Example: You roll a fair die and want to find the probability of rolling a number greater than $4$. The event is \( \{5, 6\} \), so we compute that \( P(\{5, 6\}) = \frac{2}{6} = \frac{1}{3} \). In other words, the chance that you roll a number greater than $4$ is $33.\overline{33}\%$.