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STATISTICS 200 Lecture #11 Tuesday, September 27, 2016 Textbook: Sections 7.1 through 7.4 Objectives: • Introduce notions of personal probability and relative frequency • Understand definitions of sample space, outcome, and event; identify these concepts in a simple probability experiment. • Identify complementary events and handle probability calculations • Identify mutually exclusive events and handle probability calculations • Identify independent events and handle probability calculations Last Week… 5 6 Population, Sample Census, Survey Confidence intervals • P-hat • Margin of error • Interpretation Sampling Gathering Data Observational Studies Retrospective Prospective Randomized Experiments Controls, placebos, blinding matched-pair, block, repeated-measure This week… Randomness Interpretations of probability 7 Probability • Relative Frequency • Personal Probability Sample spaces and events • Complementary • Mutually Exclusive • Dependent / independent Flawed intuition More probability practice Basic Rules • Complement rule • Addition rule • Multiplication rule Randomness The world full of random circumstances. A random circumstance is one in which the outcome is unpredictable. Examples: • • • • Outcome of coin toss (Heads or Tails) Which cards you are dealt in poker Whether it rains tomorrow What team will win the next Super Bowl. More on randomness Scientists can phrase more than you’d expect as random circumstance • • • • • • • Disease status / duration / symptoms Your DNA. Number of children per family Eye-color Result of a medical screening Time it takes to walk from here to IST building Number of people in line at Jamba Juice Probability A term we’ve probably all heard before 0 and Most generally, a number between ___ 1 ___ assigned to a possible outcome of a random circumstance. Two ways we can discuss probability: 1. Personal probability 2. Relative-frequency probability The personal probability interpretation The personal probability of an event is the degree to which a given individual believes that the event will happen. A.k.a. subjective probability, since the personal probability can change from one person to another. Examples: • • Probability a specific job candidate will be a good fit for the company Probability that life in the US will be better in 10 years. 8 Relative-Frequency Interpretation of Probability • Applies when a situation can be repeated many times ______ Relative-frequency Probability of a specific proportion outcome is defined as the ___________ of times it would occur over the long run. • can not be used to determine what the outcome will be on a single occasion _________ 9 Assign Probability: Relative Frequency Approach Method 1 assumption • make an _____________ about the physical world Method 2 • observe the relative frequency over __________repetitions many *Won’t consider Personal Probability Assignment Method 10 Example: Relative Frequency Assignment Make an Assumption about physical world Population: company sells candy with the ratio of red and blue candy shown here. Event: pick a piece of red candy equally likely to be picked Assumption: All candies are ______ 3/10 = 0.3 P(Event) = ____________ 11 Example: Relative Frequency Assignment Observe the relative frequency over many trials If we don’t know the population of candy, we can estimate the probability of red by drawing many candies. For each trial, we draw a single candy, then replace it and mix up bag before conducting another trial. This type of sampling is called Sampling with replacement . _______________________ 12 Example: Relative Frequency Assignment 0.3 red candy sample As the number of trials increases, the _________ proportion population proportion of of red candies approaches the true _________ red candies. 13 Sample Space: All possible outcomes from an experiment Experiment: Roll Two Dice 36 equally likely to occur ____________ outcomes. All are ________ 14 Simple Event: A set of a single outcome from the sample space Example: The simple event is… observe snake eyes. 1/36 Probability(snake eyes) = ________ 15 Event: Set of one or more outcomes. Example: The event is… observe a “2” on the first roll 6 outcomes in the event. Probability(event): ________ 6/36 = 1/6 ___ 16 Event: Set of one or more outcomes. Example: The event is… observe a “2” on either roll. What is the probability of this event? (A) 1/36 (B) 2/36 (C) 6/36 (D) 11/36 (E) 12/36 Again: In this example it’s reasonable to assume each outcome in the sample space is equally likely. Another example Flip two coins – a nickel and a quarter Sample space: {(HH),(HT),(TH),(TT)} {(HH)} Simple event: getting both heads _____ {(HH), (HT), (TH)} Event: Getting at least one head ______ Probability of at least one head = 3/4 _____ Note: we often use capital letters to refer to events, such as A, B, C, … 18 Definitions & Probability Rules • Event: includes outcomes that are of interest • Complement: includes outcomes are not of interest Box represents Sample space ___________ Circle represents Event A ___________ Everything outside of circle represents C Complement of event A: A’ or A _____________________ Definitions & Probability Rules Previous example of flipping two coins: Event A = getting at least one Head. {(HH), (HT),(TH)} Complement of A: Not getting any heads: {(TT)} Rule 1: Complement Rule P(A) + P(Ac) = 1, If Ac represents the complement of A So: P(Ac) = 1 – P(A) In our example: 1 – 3/4 1/4 P(Ac)= ___________ = _____ 20 Definitions & Probability Rules • Two Events are Mutually exclusive (disjoint) _________________ if with a single observation, the two events do not have shared outcomes. any _______ overlap No _________ Between the two events Definitions & Probability Rules Rule 2B: Additive Rule P(A or B)=P(A) + P(B) if events A and B are mutually exclusive. Continue coins example: Event A = get only heads Event B = get only tails A, B are mutually exclusive ½ P(A) + P(B) = _______ ¼ + ¼ = _____ P(A or B) = __________ Definitions & Probability Rules Rule 2A: Additive Rule (general) P(A or B)=P(A) + P(B) – P(A and B). A and B A B Event A: get at least one head Event B: get at least one tail {(HH),(HT),(TH),(TT)} A ¾ + ¾ - ½ = _______ 1 P(A or B) = __________ B Definitions & Probability Rules Two events are independent if knowing that one does not change the will occur (or has occurred) _______ probability that the other occurs. Two events are dependent if knowing that one will changes the probability occur (or has occurred) _______ that the other occurs. Independent is not the same as mutually exclusive! Definitions & Probability Rules Rule 3B: Multiplication Rule P(A and B) = P(A)×P(B) if Events A and B are independent. Back to our standard example…. Event A: The nickel lands heads Event B: The dime lands heads Independent events {(HH),(HT),(TH),(TT)} A B (½) x (½) = _______ P(A and B) = __________ ¼ Example Maria wants to take French or Spanish, or both. But classes are closed, ands he must apply to enroll in a language class. She has a 60% chance of being admitted to French, a 50% chance of being admitted to Spanish, and a 20% chance of being admitted to both French and Spanish. If she applies to both French and Spanish, the probability that she will be enrolled in either French or Spanish (or possibly both) is…. French 0.6 P(French) = ______ 0.5 P(Spanish) = ________ 0.2 P(French and Spanish) = ______ Spanish Example The probability that she will be enrolled in either French or Spanish (or possibly both) is…. P(French) +P(Spanish) – P(both) P(French or Spanish) = __________ 0.6 + 0.5 – 0.2 = _______ 0.9 = _____ Clicker Question: Are these events independent? A. Yes B. No Example The probability that she will be enrolled in either French or Spanish (or possibly both) is…. P(French) +P(Spanish) – P(both) P(French or Spanish) = __________ 0.6 + 0.5 – 0.2 = _______ 0.9 = _____ Clicker Question: Are these events mutually exclusive? A. Yes B. No 28 Summary of Rules Rule 1: Complement Rule P(A) + P(Ac) = 1 if Ac represents the complement of A Rule 2B: Additive Rule P(A or B) = P(A) + P(B) if Events A and B are mutually exclusive Note: two events that are complements are always mutually exclusive Rule 3B: Multiplication Rule P(A and B) = P(A)×P(B) if Events A and B are independent If you understand today’s lecture… • 7.9, 7.11, 7.17, 7.23, 7.25, 7.29, 7.33, 7.39, 7.41, 7.43, 7.45, 7.57, 7.59 Objectives: • Introduce notions of personal probability and relative frequency • Understand definitions of sample space, outcome, and event; identify these concepts in a simple probability experiment. • Identify complementary events and handle probability calculations • Identify mutually exclusive events and handle probability calculations • Identify independent events and handle probability calculations