Math 311

Class 2

Sample Space and Events

  • Experiment

    • Outcomes
    • Sample space: the set of all possible outcomes
    • Event: a subset of the sample space

  • Disjoint or mutually exclusive events

  • Intersection of events: \(A\) and \(B\) (both \(A\) and \(B\) happen)

    \(A \cap B\), or simply \(AB\)

  • Union of events: \(A\) or \(B\) (at least one of \(A\) and \(B\) happen)

    \(A \cup B\), or (rarely) \(A + B\)

  • Complement of an event \(A\): the set of all the outcomes that are not in \(A\)

  • Atomic (or elementary) event: contains a single outcome

Probability Measure

  • Observable events
  • Probability: a function from the set of all observable events to \([0,1]\)
    • \(0 \le \operatorname{P}(A) \le 1\)
    • \(\operatorname{P}(S) = 1\)
    • For any sequence of events \(A_i\), \(i = 1, 2, 3 \ldots\) that are mutually exclusive, \(\displaystyle\operatorname{P}\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \operatorname{P}\left(A_i\right)\)
  • Example: finite events
  • Example: Probability of \(\overline{A}\)
  • Addition formula
  • Inclusion-exclusion principle

Example

Rolling two fair dice

  • What is the probability of a pair?
  • Suppose we know that the sum was 4, what is the probability of a pair?
  • Suppose we know that the sum was 5, what is the probability of a pair?
  • Suppose we know that the sum was 6, what is the probability of a pair?

Another Example

A photon strikes a square sensor.

  • What is the probability that the collision will happen in the top triangle?
  • What is the probability that the collision will happen in the top triangle, given it happens in the upper half?
  • What is the probability that the collision will happen in the left triangle?
  • What is the probability that the collision will happen in the left triangle, given it happens in the upper half?

Conditional Probability

  • \(\operatorname{P}(B \mid A) = {}\) the conditional probability of \(B\), given that \(A\) happened.

  • \(\displaystyle\operatorname{P}(B\mid A) = \frac{\operatorname{P}(AB)}{\operatorname{P}(A)}\)

  • Multiplication rule: \(\operatorname{P}(AB) = \operatorname{P}(A)\times \operatorname{P}(B \mid A)\)

  • Example: We have an urn with 5 tokens, two red and three blue. We remove two of the tokens. What is the probability that both are blue?

  • Example: We have an urn with 5 tokens, two red and three blue. We remove one of the tokens, record its color, and return it back. Then we again remove a token. What is the probability that both tokens are blue?

Independence

We saw that sometimes \(\operatorname{P}(B\mid A) = \operatorname{P}(B)\).

We say that \(B\) is independent od \(A\).

Theorem: If \(\operatorname{P}(A) \neq 0\) and \(\operatorname{P}(B)\neq 0\), then the following are equivalent:

  1. \(\operatorname{P}(A\mid B) = \operatorname{P}(A)\)
  2. \(\operatorname{P}(B\mid A) = \operatorname{P}(B)\)
  3. \(\operatorname{P}(AB) = \operatorname{P}(A)\operatorname{P}(B)\)

Definition: We say that \(A\) and \(B\) are independent events if \(\operatorname{P}(AB) = \operatorname{P}(A)\operatorname{P}(B)\)

Definition: Events \(A_1\), \(A_n\), \(A_3\), …, \(A_n\) are jointly independent if for every subset \(J\) of \(\left\{1, 2, 3, \ldots, n\right\}\),

\[ \operatorname{P}\left(\bigcap_{j \in J} A_j\right) = \prod_{j\in J} \operatorname{P}\left(A_j\right) \]