University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

447

Pages:

4
Author:

Niklas

Lecture 13 Probability helps in avoiding randomness disguised and perceived as non-randomness. Frequentist Interpretation: Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. Example: Flipping a coin 500 times, the probability of heads approaches 50% as the number of flips increases. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. Random experiment: A process leading to two or more possible outcomes without knowing exactly which one will occur. Sample space (S): The set of all possible basic outcomes. Event (A): An outcome or a set of possible outcomes. Relative frequency probability: P(A) = nA/n, where nA is the number of times event A occurs and n is the total number of trials. Probability The classical probability is the proportion of times that an event will occur, assuming that all outcomes in a sample space are equally likely to occur. P(A) = Na/N Postulates: 0 ≤ P(A) ≤ 1 P(S) = Σ P(Oi) = 1 Complement rule: P(A') = 1 - P(A) Additional rule of probability: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Venn Diagrams: Two overlapping circles represent events A and B. The intersection of the circles represents the event A ∩ B. The union of the circles represents the event A ∪ B. Random experiment: rolling a dice Sample space: S = {1, 2, 3, 4, 5, 6} Event A: getting a 6 Event B: getting an even number P(A) = 1/6 = 17% P(B) = 1/2 = 50% P(A ∩ B) = 1/6 = 17% P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 1/6 + 1/2 - 1/6 = 1/2 = 50% Random experiment: tossing a coin 3 times Sample space: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Event A: two heads Event B: more than one head P(A) = 3/8 = 37.5% P(B) = 4/8 = 1/2 = 50% A ⊂ B, so P(A ∩ B) = P(A) = 3/8 P(A ∪ B) = P(B) = 1/2 = 50% Random experiment: roulette Sample space: S = {0, 1, ..., 36} Event A: first column Event B: red Event C: 0 A ∩ B = {1, 7, 16, 19, 25, 34} A ∪ B = {1, 2, 3, 4, 5, 7, 9, 10, 12, ..., 34} A ∩ C = Ø A ∪ C = {0, 1, 4, 7, ..., 34} C̄ = S - C = {1, 2, 3, ..., 36} P(A ∩ B) = 6/37 P(A ∪ B) = 24/37 P(A ∩ C) = Ø P(A ∪ C) = 13/37 P(C̄) = 1 - P(C) = 36/37 Exercise 1 Canada's official languages: English and French Survey: A survey asks Canadians about their mother tongue. Data: A table showing the distribution of mother tongues (English, French, Asian/Pacific, Other). Calculations: a) Complete the table: ΣP = 1 => 0.59 + 0.23 + 0.07 + x = 1 => x = 0.11 b) Probability of mother tongue not English: P(not English) = 1 - P(English) = P(French) + P(Asian/Pacific) + P(Other) = 1 - 0.59 = 0.41 Exercise 2 Scenario: A survey about whether people want text messaging on their phones. Data: A table showing the probabilities of different preferences (text messaging only, phone calls only, both, neither). Calculations: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.75 + 0.10 - 0.06 = 0.79