Section Constructing Models of Random Behavior Objectives: 1.Build probability models by observing data 2.Build probability models by constructing a sample space of equally likely outcomes (symmetry) 3.See how the Law of Large Numbers relates data to probability

Section Constructing Models of Random Behavior Fundamental Facts About Probability An event A is a set of possible outcomes from a random situation. Probability is a number between 0 and 1 (or between 0% and 100%) that tells how likely it is for an event to happen. Events that can’t happen have probability 0. Events that are certain to happen have probability 1. The probability that event A happens is denoted P(A); the probability that event A doesn’t happen is denoted P(A’) = 1 - P(A). The event A’ is called the complement of A.

Section Constructing Models of Random Behavior Fundamental Facts About Probability Example: Rolling a six-sided die. Outcomes: {1, 2, 3, 4, 5, 6} Event: “rolling a even number” {2, 4, 6} Probabilities: P(1) = … = P(6) = 1/6 P(even) = 1/2 P(1 or 2 or 3 or 4 or 5 or 6) = 1 (certain event) P(7) = 0 (impossible event) P(2 or 3 or 4 or 5 or 6) = 1 - P(1) = 5/6 (complement)

Section Constructing Models of Random Behavior Fundamental Facts About Probability If you have a list of all possible outcomes and all outcomes are equally likely, then the probability of a specific outcome is and the probability of an event is

Section Constructing Models of Random Behavior Fundamental Facts About Probability Example: Rolling a six-sided die All possible outcomes: {1, 2, 3, 4, 5, 6} Let A be the event “rolling an even number”

Section Constructing Models of Random Behavior Probability Distributions A probability distribution gives all possible values resulting from a random process and the probability of each. Example: Flip a “fair coin” twice. What is the probability of 0, 1, or 2 heads? Outcomes: HH HT TH TT P(HH) = P(HT) = P(TH) = P(TT) = 1/4 The probability distribution corresponding to the random process of flipping a fair coin twice is: Number of HeadsProbability 01/4 11/2 21/4

Section Constructing Models of Random Behavior Where Do Probabilities Come From? Observed data (long-run relative frequencies) Observation of thousands of births has shown that about 51% of newborns are boys. P(boy) ≈ 0.51 Symmetry (equally likely outcomes) Flipping a coin. Symmetry suggests that heads and tails are equally likely. P(heads) = P(tails) = 0.5 Subjective estimates (may be based on data) What is the probability that Tom will be accepted into his first-choice college?

Section Constructing Models of Random Behavior Sample Spaces A sample space for a chance process is a complete list of disjoint outcomes. All of the outcomes in a sample space must have a total probability equal to 1. Disjoint means that two different outcomes can’t occur on the same opportunity. The term mutually exclusive is sometimes used instead of disjoint.

Section Constructing Models of Random Behavior Sample Spaces Example: Rolling a six-sided die. Sample space (a complete list of disjoint outcomes) {1, 2, 3, 4, 5, 6} {odd, even}

Section Constructing Models of Random Behavior Data and Symmetry How can you tell if the outcomes in your sample space are equally likely? Compare your model’s predictions with the actual results to see if you have a good fit. Example: Rolling a six-sided die The only thing that makes one side different from another is the number of dots It seems unlikely that the number of dots would have much of an effect on the probability. Verify by rolling the die many times. See if the actual results match the model.

Section Constructing Models of Random Behavior Activity 5.1a: Spinning Pennies

Section Constructing Models of Random Behavior The Law of Large Numbers In random sampling, the larger the sample, the closer the proportion of successes in the sample tends to be to the proportion in the population. The difference between a sample proportion and the population proportion must get smaller as the sample size gets larger.

Section Constructing Models of Random Behavior The Law of Large Numbers Example: Fifty Fathoms demos Law of Large Numbers and Law of Large Numbers 2

Section Constructing Models of Random Behavior The Fundamental Principle of Counting For a two-stage process with n 1 possible outcomes for stage 1 and n 2 possible outcomes for stage 2, the number of possible outcomes for the two stages taken together is n 1 n 2. More generally, if there are k stages, with n i possible outcomes for stage i, then the number of possible outcomes for all k stages taken together is n 1 n 2 …n k.

Section Constructing Models of Random Behavior Tree Diagrams Example: A tree diagram of all possible outcomes when flipping a fair coin twice.

Section Constructing Models of Random Behavior Two-way Tables Example: There are 36 equally likely outcomes when rolling two dice. Second Roll First Roll ,11,21,31,41,51,6 22,12,22,32,42,52,6 33,13,23,33,45,33,6 44,14,24,34,45,44,6 55,15,25,35,45,55,6 66,16,26,36,46,56,6

Section Constructing Models of Random Behavior Summary A probability model is a sample space together with an assignment of probabilities. The sample space is a complete list of disjoint outcomes where Each outcome is assigned a probability between 0 and 1 The sum of all the probabilities is 1.

Section Constructing Models of Random Behavior Summary The probability of an event is the number of outcomes that make up the event divided by the total number of possible outcomes.

Section Constructing Models of Random Behavior Summary The main practical application of equally likely outcomes are in the study of random samples and in randomized experiments. In a survey, all possible simple random samples are equally likely. In a completely randomized experiment, all possible assignments of treatments to units are equally likely.

Section Constructing Models of Random Behavior Summary The only way to decide whether a probability model is a reasonable fit to a real situation is to compare probabilities derived from the model with probabilities estimated from observed data.

Section Constructing Models of Random Behavior Summary The Fundamental Principle of Counting If you have a process consisting of k stages with n i outcomes for stage i, the number of outcomes for all k stages taken together is n 1 n 2 n 3 · · · n k