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# Are the following valid sample spaces 1 2 3 4 6

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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 4-sided 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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