A number that encodes our knowledge or belief about the collective “likelihood” of the elements of the event.
\(\text{P}(A)\): the probability of event \(A\).
Roll a fair die 🎲. Which of the two would you bet on?
Roll two dice 🎲 🎲. Which of the two would you bet on?
If \(A_1, A_2, A_3, \cdots\) is an infinite collection of disjoint events, then
\[ \text{P}(A_1 \cup A_2 \cup A_3 \cup \cdots) = \text{P}(A_1) + \text{P}(A_2) + \text{P}(A_3) + \cdots\]
Recall the non-negativity axiom:
\[\text{P}(A)\geq0,\; \text{for any event $A$}\]
Do we need to also add the following as an axiom?
\[\text{P}(A) \leq 1, \;\text{for any event } A\]
\[\text{P}(A) + \text{P}(A^c) = 1, \;\text{for any event } A\]
\[ \text{P}(A \cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B), \]
\[\text{for any events $A$ and $B$}\]
Assumptions
We randomly select a student. What is the probability that