Probability, Statistics

Probability

Probability

We will call the probability of a \Omega sample space to any application that complies:

\begin{cases} \Omega \rightarrow R \\ \omega \rightarrow p(\omega) \in \left[0, 1\right] \end{cases}


where the value between 0 and 1 tries to quantify the possibility of that event occurring. It is also usually measured in percentage, so a probability of 1 equals 100% and one of 0 to 0%

  1. P(A) \ge 0, \forall A \text{ event}
  2. P( \Omega) = 1
  3. P(A \cup B) = P(A) + P(B)\text{ si }A \cap B = \emptyset

Properties

  1. P(A) \le 1
  2. P(\emptyset) = 0
  3. P(A^c) = 1 - P(A)
  4. Si B \subset A \Rightarrow P(A - B) = P(A) - P(B)
  5. P(A - B) = P(A) - P(A \cap B)
  6. P(A \cup B) = P(A) + P(B) - P(A \cap B)
  7. P(A_1 \cup \cdots \cup A_n) = P(A_1) + \cdots + P(A_n); \text{ Si } A_i \cap A_j = \emptyset; \forall \not= j
  8. P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C)- P(B \cap C) + P(A \cap B \cap C)
  9. P(A \cup B \cup C \cup D) = P(A) + P(B) + P(C) + P(D) - P(A \cap B) - P(A \cap C) - P(A \cap D) - P(B \cap D) - P(C \cap D) + P(A \cap B \cap C \cap D) + P(B \cap C \cap D) - P(A \cap B \cap C \cap D)

Rule of addition

The addition rule or sum rule states that the probability of occurrence of any particular event is equal to the sum of individual probabilities, if the events are mutually exclusive, i.e. two cannot occur at the same time

P(A) \cup P(B) = P(A) + P(B) if A and B are mutually exclusive

P(A\cup B) = P(A) + P(B) - P(A\cap B) if A and B are not mutually exclusive

Being:

\scriptsize\begin{cases}\text{P(A) = probability of occurrence of the event A}\\ \text{P(B) = probability of occurrence of the event B}\\ P(A \cap B)\text{ = probability of simultaneous occurrence of events A y B}\end{cases}

Rule of multiplication

The rule of multiplication states that the probability of occurrence of two or more events are statistically independent is equal to the product of their probabilities, individual

P(A \cap B) = P(A\cdot B) = P(A) \cdot P(B) if A and B are independent

P(A \cap B) = P (A \cdot B) = P(A)\cdot P(B|A) if A and B are dependent

Being P(B|A) the probability that B will occur having given or verified event A

Rule of Laplace

It \Omega sample space where the sample points have the same possibility of occurrence, To event, then:

P(A) = \frac{n^{\underline{0}}\text{ of favorable cases}}{n^{\underline{0}}\text{ of possible cases}}

Frequency Probability (Von Mises)

It \Omega sample space associated with a random phenomenon, be A event. The frequency probability of A occurring is the relative frequency of the number of times it occurs when we repeat the random phenomenon \infty times

\lim\limits_{n\to\infty} \frac{n^{\underline{0}}\text{ of times it happens}}{n}