Unit 2: Probability and distributions Lecture 1: Probability and conditional probability Statistics 101 Thomas Leininger May 21, 2013
Announcements Announcements 1 Probability 2 Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap Marginal, joint, conditional 3 Statistics 101 U2 - L1: Probability Thomas Leininger
Announcements Announcements PS #1 due today PS #2 assigned (due Friday) Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 2 / 30
Announcements Visualization of the day http://www.washingtonpost.com/blogs/the-fix/wp/2013/01/22/why-republicans-should-stop-talking-about-roe-v-wade/ Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 3 / 30
Announcements Visualization of the day wrong, overturn wrong, don't overturn not wrong, overturn not wrong, don't overturn NA 0 10 20 30 40 http://www.washingtonpost.com/blogs/the-fix/wp/2013/01/22/why-republicans-should-stop-talking-about-roe-v-wade/ Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 3 / 30
Probability Announcements 1 Probability 2 Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap Marginal, joint, conditional 3 Statistics 101 U2 - L1: Probability Thomas Leininger
Probability Randomness Announcements 1 Probability 2 Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap Marginal, joint, conditional 3 Statistics 101 U2 - L1: Probability Thomas Leininger
Probability Randomness Random processes A random process is a situation in which we know what outcomes could happen, but we don’t know which particular outcome will happen. Examples: coin tosses, die rolls, iTunes shuffle, whether the stock market goes up or down tomorrow, etc. It can be helpful to model a process as random even if it http://www.cnet.com.au/ is not truly random. itunes-just-how-random-is-random-339274094.htm Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 4 / 30
Probability Defining probability Announcements 1 Probability 2 Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap Marginal, joint, conditional 3 Statistics 101 U2 - L1: Probability Thomas Leininger
Probability Defining probability Probability There are several possible interpretations of probability but they (almost) completely agree on the mathematical rules probability must follow. P ( A ) = Probability of event A 0 ≤ P ( A ) ≤ 1 Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 5 / 30
Probability Defining probability Probability There are several possible interpretations of probability but they (almost) completely agree on the mathematical rules probability must follow. P ( A ) = Probability of event A 0 ≤ P ( A ) ≤ 1 Frequentist interpretation: The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an infinite number of times. Single main stream school until recently. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 5 / 30
Probability Defining probability Probability There are several possible interpretations of probability but they (almost) completely agree on the mathematical rules probability must follow. P ( A ) = Probability of event A 0 ≤ P ( A ) ≤ 1 Frequentist interpretation: The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an infinite number of times. Single main stream school until recently. Bayesian interpretation: A Bayesian interprets probability as a subjective degree of belief: For the same event, two separate people could have differing probabilities. Largely popularized by revolutionary advance in computational technology and methods during the last twenty years. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 5 / 30
Probability Law of large numbers Announcements 1 Probability 2 Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap Marginal, joint, conditional 3 Statistics 101 U2 - L1: Probability Thomas Leininger
Probability Law of large numbers Question Which of the following events would you be most surprised by? (a) 3 heads in 10 coin flips (b) 3 heads in 100 coin flips (c) 3 heads in 1000 coin flips Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 6 / 30
Probability Law of large numbers Question Which of the following events would you be most surprised by? (a) 3 heads in 10 coin flips (b) 3 heads in 100 coin flips (c) 3 heads in 1000 coin flips Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 6 / 30
Probability Law of large numbers Law of large numbers Law of large numbers states that as more observations are collected, the proportion of occurrences with a particular outcome, ˆ p n , converges to the probability of that outcome, p . Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 7 / 30
Probability Law of large numbers Law of large numbers (cont.) When tossing a fair coin, if heads comes up on each of the first 10 tosses, what do you think the chance is that another head will come up on the next toss? 0.5, less than 0.5, or more than 0.5? H H H H H H H H H H ? Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 8 / 30
Probability Law of large numbers Law of large numbers (cont.) When tossing a fair coin, if heads comes up on each of the first 10 tosses, what do you think the chance is that another head will come up on the next toss? 0.5, less than 0.5, or more than 0.5? H H H H H H H H H H ? The probability is still 0.5, or there is still a 50% chance that another head will come up on the next toss. P ( H on 11 th toss ) = P ( T on 11 th toss ) = 0 . 5 Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 8 / 30
Probability Law of large numbers Law of large numbers (cont.) When tossing a fair coin, if heads comes up on each of the first 10 tosses, what do you think the chance is that another head will come up on the next toss? 0.5, less than 0.5, or more than 0.5? H H H H H H H H H H ? The probability is still 0.5, or there is still a 50% chance that another head will come up on the next toss. P ( H on 11 th toss ) = P ( T on 11 th toss ) = 0 . 5 The coin is not due for a tail . Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 8 / 30
Probability Law of large numbers Law of large numbers (cont.) When tossing a fair coin, if heads comes up on each of the first 10 tosses, what do you think the chance is that another head will come up on the next toss? 0.5, less than 0.5, or more than 0.5? H H H H H H H H H H ? The probability is still 0.5, or there is still a 50% chance that another head will come up on the next toss. P ( H on 11 th toss ) = P ( T on 11 th toss ) = 0 . 5 The coin is not due for a tail . The common (mis)understanding of the LLN is that random processes are supposed to compensate for whatever happened in the past; this is just not true and is also called gambler’s fallacy (or law of averages ). Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 8 / 30
Probability Disjoint and non-disjoint outcomes Announcements 1 Probability 2 Randomness Defining probability Law of large numbers Disjoint and non-disjoint outcomes Probability distributions Independence Recap Marginal, joint, conditional 3 Statistics 101 U2 - L1: Probability Thomas Leininger
Probability Disjoint and non-disjoint outcomes Disjoint and non-disjoint outcomes Disjoint (mutually exclusive) outcomes: Cannot happen at the same time. The outcome of a single coin toss cannot be a head and a tail. A student cannot fail and pass a class. A card drawn from a deck cannot be an ace and a queen. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 9 / 30
Probability Disjoint and non-disjoint outcomes Disjoint and non-disjoint outcomes Disjoint (mutually exclusive) outcomes: Cannot happen at the same time. The outcome of a single coin toss cannot be a head and a tail. A student cannot fail and pass a class. A card drawn from a deck cannot be an ace and a queen. Non-disjoint outcomes: Can happen at the same time. A student can get an A in Stats and A in Econ in the same semester. Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 9 / 30
Probability Disjoint and non-disjoint outcomes Union of non-disjoint events What is the probability of drawing a jack or a red card from a well shuffled full deck? Figure from http://www.milefoot.com/math/discrete/counting/cardfreq.htm . Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 10 / 30
Probability Disjoint and non-disjoint outcomes Union of non-disjoint events What is the probability of drawing a jack or a red card from a well shuffled full deck? P ( jack or red ) = P ( jack ) + P ( red ) − P ( jack and red ) = 4 52 + 26 52 − 2 52 = 28 52 Figure from http://www.milefoot.com/math/discrete/counting/cardfreq.htm . Statistics 101 (Thomas Leininger) U2 - L1: Probability May 21, 2013 10 / 30
Recommend
More recommend
