Unformatted text preview: question interested in) Consider the experiment of throwing a die once. Are the following
valid sample spaces?
Ω = {1, 2, 3, 4, 6}
Ω = {{1 or 3}, {2}, {3}, {4}, {5}, {6}}
Ω = {1, 2, 3, 4, 5, 6}
Ω = {{1 or 3 or 5}, {2 or 4 or 6}} M. Chen ([email protected]) ENGG2430C lecture 2 8 / 23 Sample Space at the Right Level of Granularity Ask an unknown person Alice about her birthday
Sample space #1: Ω = {Jan 1, Jan 2, . . . , Dec 30, Dec 31}
Sample space #2: Ω = {Jan, Feb, . . . , Nov, Dec} What is the probability that Alice was born in April? M. Chen ([email protected]) ENGG2430C lecture 2 9 / 23 Sample Space with Sequential Descriptions Neat for experiments with an inherently sequential character
Receiving seven successive packets over a WiFi link
Rolling a 4sided die twice (Fig. 1.3 in the textbook)
M. Chen ([email protected]) ENGG2430C lecture 2 10 / 23 Probability Model: Probability Law Event: a subset of the sample space Ω
Probabilities are assigned to events:
For any event A ⊆ Ω, the probability law speciﬁes P (A) Axioms
P (A) ≥ 0, for all subset A in Ω
P (Ω) = 1
If A1 , A2 , A3 , · · · are disjoint events (i.e., Ai ∩ Aj = 0), then:
/
P (A1 ∪ A2 ∪ A3 ∪ · · · ) = P (A1 ) + P (A2 ) + P (A3 ) + · · ·
Question: P (0) =? Answer: 0
/ M. Chen ([email protected]) ENGG2430C lecture 2 11 / 23 An Example: Discrete Uniform Law Let all outcomes be equally likely
Then,
P (A) = number of outcomes in A
total number of outcomes in Ω Just count... M. Chen ([email protected]) ENGG2430C lecture 2 12 / 23 The Birthday Example
Ask an unknown person Alice : what date is your birthday?
Let every possible outcome have probability 1
365 Event A ={Alice was born in January}
P (A) = 31
365 Ask two unknown persons Alice and Bob: what dates are your birthdays?
Let every possible outcome have probability 1
3652 Event A ={both Alice and Bob were born in January}
P (A) = 31
365 31
× 365 M. Chen ([email protected]) ENGG2430C lecture 2 13 / 23 Properties of Probability Law Exercise: consider a probability law P , and let A, B , and C be events.
Prove the following using only the axioms. (Hint: construct disjoint
sets and apply Axiom #3)
If A ⊂ B , then P (A) ≤ P (B ) (logic: construct two disjoint sets A and
C so that A ∪ C = B ; then apply Axiom #...
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 Spring '14
 Probability, Probability theory, Birthday problem, Minghua Chen

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