Note: (*) means that bullet point is not mentioned in class but in handout.
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If there are N equally likely possibilities, of which one must occur and n are regarded as favorable, or as a “success”, then the probability of a “success” is given by the ratio $ \frac{n}{N} $.
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Experiment: Any process of observation or measurement.
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Outcomes: The results one obtains from an experiment. (Ex. sensor readings, value counts, “yes”, or “no” answers)
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Sample space: The set of all possible outcomes of an experiment, denoted by $S$.
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Element/Sample Point: Each outcome in a sample space.
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Discrete sample space: A sample space that contains either a finite or a countable number of distinct sample points.
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Event: A subset of a sample space.
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Union $A \cup B$: the subset of S that contains all the elements that are either in A or in B, or in both.
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Intersection $A \cap B$: the subset of S that contains all the elements that are both in A and B.
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Complement $A’$ of $A$: the subset of S that contains all the elements of S that are not in A.
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Events A and B are mutually exclusive if $A \cap B = \emptyset$ (A and B doesn’t occur at the same time).
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A is contained in B is written as $A \subset B$.
- Basic operation properties:
- Commutative
- $A \cup B = B \cup A$
- $A \cap B = B \cap A$
- Associative
- $(A \cup B) \cup C = A \cup (B \cup C)$
- $(A \cap B) \cap C = A \cap (B \cap C)$
- De Morgan’s Law:
- $(A \cup B)’ = A’ \cap B’$
- $(A \cap B)’ = A’ \cup B’$
- $(A \cup B \cup C)’ = A’ \cup B’ \cup C’$
- $(A \cap B \cap C)’ = A’ \cap B’ \cap C’$
- Commutative
- Probability Axioms: given $P(A)$ the probability of event A,
- $P(A)$ is a nonnegative real number; that is, $P(A) \geq 0 \text{ for any } A \subseteq S$.
- $P(S) = 1$
- If $A_1$, $A_2$, …, is a finite or infinite sequence of mutulally exclusive events of S, then $P(A_1 \cup A_2 \cup A_3 \cup …) = P(A_1) + P(A_2) + P(A_3) + …$
- Theorem 2.1: If:
A is an event in S, and
S is a discrete sample space, then
P(A) is the sum of the probabilities of the inidividual outcomes in A.
- Rules of Probability:
- If A and A’ are complementary events in a sample space S, then $P(A’) = 1 - P(A)$.
- $P(\emptyset) = 0$ for any sample space S.
- If A and B are events in a sample space S and $A \subset B$, then $P(A) \leq P(B)$.
- $0 \leq P(A) \leq 1$ for any event A.
- If A and B are any events in a sample space S, then \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
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If A, B and C are events in a sample space S,then \(P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)\)
- Odds of an event A: P(A) to P(A’), provided neither probability is zero.
- Odds are usually written in terms of positive integers having no common factor.
- If the odds are a to b that an event A will occur, then $P(A) = \frac{a}{a + b}$.
- Conditional Probability:
- If A and B are any 2 events in a sample space S and P(A) $\neq$ 0, the conditional probability of B given A is $P(B \mid A) = \frac{P(A \cap B)}{P(A)}$.
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If A and B are events in a sample space S and P(A) $\neq$ 0, then $P(A \cap B) = P(B) \times P(A \mid B) = P(A) \times P(B \mid A)$.
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Events A and B are independent if and only if $P(A \cap B) = P(A)P(B) \Leftrightarrow P(A \mid B) = P(A) \Leftrightarrow P(B \mid A) = P(B)$.
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The rule of total probability: If the events $B_1$, $B_2$, …, and $B_k$ constitute a partition of the sample space S and $P(B_i) \neq 0$, for i = 1, 2, …, k, then for any event A in S: $P(A) = \sum_{i=1}^{k}P(B_i)P(A \mid B_i)$.
- Bayes’ Theorem: If $B_1$, $B_2$, …, and $B_k$ constitute apartition of the sample space S and $P(B_i) \neq 0$ for i=1,2, …, k then for any event A such that P(A) $\neq$ 0, \(P(B_r \mid A) = \frac{P(B_r)P(A \mid B_r)}{\sum_{i=1}^{k}P(B_i)P(A \mid B_i)}\)
- When $S = B \cup B’$, $P(B \mid A) = \frac{P(B)P(A \mid B)}{P(B)P(A \mid B) + P(B’)P(A \mid B’)}$
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(*)If A, B, and C are events in a sample space S and $P(A \cap B) \neq 0$, then $P(A \cap B \cap C) = P(B) \times P(A \mid B) \times P(C \mid A \cap B)$.
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(*)If A and B are independent, then A and B’ are also independent: $P(A \cap B’) = P(A) \times P(B’)$.
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(*)Events $A_1$, $A_2$, …, and $A_k$ are independent if and only if the probability of the intersection of any 2, 3, …, k of these events equals the product of their respective probabilities.
- (*)Note that 3 or more events can be pairwise independent without being independent.
Questions
- What is the definition of a continuous sample space, according to the handhout?