Basic Concepts

A few more rules can be inferred from the three axioms:

4. \( P(\emptyset) = 0 \)
5. \( P(A^c) = 1 - P(A) \) [$A^c$ is the complement of $A$.]
6. If $A \subseteq B$, then $P(A) \le P(B)$

Examples:

• The probability of rolling a $7$ on a die is $0$ since rolling a $7$ is a non-existent (impossible) outcome.
• The probability of rolling anything but $4$ on a die is $\frac{5}{6}$ since rolling a $4$ has $P[\text{rolling a 4}] = \frac{1}{6}$, so $P[\text{not rolling a 4}] = 1 - P[\text{rolling a 4}] = 1 - \frac{1}{6} = \frac{5}{6}$.
• The event of "rolling an even integer" is a subset of the event "rolling a number bigger than $1$" (why so?). Indeed, we have $P[\text{rolling an even integer}] = \frac{3}{6} = \frac{1}{2}$ and $P[\text{rolling a number bigger than 1}] = \frac{5}{6}$, and that $\frac{3}{6} < \frac{5}{6}$.