Chapter 14 Chapter14 Presentation 1213 From Randomness to Probability Copyright 2009 Pearson Education Inc 1 Gambling The Birth of Statistics If I flipped a coin and got 4 heads in a row would you bet money that tails will be next because it s due bet money that heads will be next because the coin is on a streak conclude that my coin is unfair not yielding 50 50 results How unusual is 4 heads in a row Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 2 Dealing with Random Phenomena What do we mean by a random phenomenon In general each occasion upon which we observe a random phenomenon is called a trial At each trial we note the value of the random phenomenon and call it an outcome When we combine outcomes the resulting combination is an event The collection of all possible outcomes is called the sample space Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 3 The Law of Large Numbers First a definition When thinking about what happens with combinations of outcomes i e events things are simplified if the individual trials are independent Roughly speaking this means that the outcome of one trial doesn t influence or change the outcome of another For example coin flips are independent Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 4 The Law of Large Numbers cont The Law of Large Numbers LLN says that the long run relative frequency of repeated independent events gets closer and closer to a single value We call the single value the probability of the event Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 5 The Law of Large Numbers Example Probability of green light looks to be about 0 30 The overall percentage of times the light is green settles down as you see more outcomes Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 6 The Law of Large Numbers vs the Law of Averages The Law of Large Numbers says relative frequencies even out only in the long run and this long run is really long infinitely long in fact The Law of Averages says for example that if you flip a fair coin 4 times and get heads all 4 times tails are due on the 5th flip If it s really a fair coin is this true Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 7 Law of Large Numbers and Coin Flips Careful the law of large numbers talks about the relative frequency of an event not the number of times the event happens Notice that the difference in the of H vs T top picture is not approaching 0 as the number of tosses increases but the relative frequency of H bottom picture is approaching 0 50 Chapter14 Presentation 1213 Relative Frequency of Heads Copyright 2009 Pearson Education Inc 8 Mathematically Determined Probabilities In some situations repeated observation is not necessary to know the probability of an event The long run relative frequency can be mathematically determined All gambling games of chance are examples Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 9 Personal Probability In everyday speech when we express a degree of uncertainty without basing it on long run relative frequencies or mathematical calculations we are stating subjective or personal probabilities Personal probabilities don t display the kind of consistency that we will need probabilities to have so we ll stick with formally defined probabilities Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 10 Modeling Probability Mathematically calculating the probability of an event is simplified when all the possible outcomes are equally likely However keep in mind that all possible outcomes are not always equally likely Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 11 Modeling Probability cont If each possible outcome is equally likely the probability of an event is the number of outcomes in the event divided by the total number of possible outcomes P A Chapter14 Presentation 1213 of outcomes in A of possible outcomes Copyright 2009 Pearson Education Inc 12 Modeling Probability Example Flip a fair coin 4 times There are 16 possible equally likely outcomes If event A is getting at least 3 heads what is P A How would you interpret this probability Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 13 Formal Probability 1 Requirement for a probability A probability is a number between 0 and 1 For any event A 0 P A 1 Chapter14 Presentation 1213 If P A 0 event A can never happen If P A 1 event A will happen with absolute certainty Copyright 2009 Pearson Education Inc 14 Formal Probability cont 2 Probability Assignment Rule Let S represents the set of all possible outcomes P S 1 Example roll a fair six sided die record the number of dots on the up face S 1 2 3 4 5 6 P S probability of getting a 1 or a 2 or a 3 or a 4 or a 5 or a 6 1 00 Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 15 Formal Probability cont 3 Complement Rule The set of outcomes that are not in the event A is called the complement of A The probability of an event occurring is 1 minus the probability that it doesn t occur P A 1 P not A Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 16 Compliment Rule cont Examples where it s easier to calculate the probability of the event not happening What s the probability that in a room of 50 people one or more people share the same birthday What s the probability at least one other person on board a plane is a certified pilot if the original and copilot dies Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 17 Formal Probability cont 4 Addition Rule Events that have no outcomes in common and thus cannot occur simultaneously are called disjoint or mutually exclusive Chapter14 Presentation 1213 Example Roll a single die Event A roll a 5 or greater Event B roll a 2 or less Event C roll an even number Events A and B are disjoint they can t happen on a single roll Events A and C are not disjoint they could both happen on a single roll when you roll a 6 Copyright 2009 Pearson Education Inc 18 Formal Probability cont 4 Addition Rule cont For two disjoint events A and B the probability that one or the other occurs is the sum of the probabilities of the two events P A or B P A P B provided that A and B are disjoint Chapter14 Presentation 1213 Copyright 2009 Pearson Education Inc 19 Addition Rule Example Recall event A is getting 3 or more heads Event B is getting an even number of tails not including zero tails Are events A and B disjoint P A P B P A
View Full Document
Unlocking...