# Section 5.1 Constructing Models of Random Behavior.

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Section 5.1 Constructing Models of Random Behavior

 Main Concepts:  List all possible outcomes of a chance process in a systematic way  Design simulations and use them to estimate probabilities  Use the Addition rule to compute probability that event A or event B (or both) occurs  Use the Multiplication rule to compute probability that event A and event B both occur.  Compute Conditional probabilities, the probability that event B occurs given that event A occurs (the most difficult)

 An event is a set of possible outcomes from a random situation: rolling dice, drawing a card, pulling a marble from a bag, result of some type of spinner…  Probability is a number between 0 and 1 (or between 0% and 100%.  Something that is certain to occur has a probability of 1.  Something that will not occur has a probability of 0.

 The probability that event A occurs is denoted by P(A).  The probability that event A does not occur is then, P(not A) = 1 – P(A)  “not A” is also called the compliment of A.  If you have a list of all possible outcomes and all outcomes are equally likely, the probability of a specific outcome is: the number of outcomes of that event / the total number of equally likely outcomes

 Page 288: Identifying Tap or Bottled water:  Probability Distribution: a chart or graph showing the possible values from the random process and the probabilities of each.  The sum of the probabilities must be….?

 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: two different outcomes can’t occur in the same opportunity. (AKA: mutually exclusive)

 Think of a Tree Diagram which maps out the possibilities.Tree Diagram  If you have n 1 possible outcomes for stage 1 and n 2 possible outcomes for stage 2, n 3 possible for stage 3 and so on, then the total possible for the stages together is n 1 n 2 n 3.  2 people guessing T or B.  3 people guessing T or B.

 6 possibilities on each. n 1 =6, and n 2 =6  So there are 6*6 = 36 equally likely outcomes.  Notice for example 3,4 and 4,3 have the same result, but were different in how they happened, so they are considered two different outcomes.  However 3,3 is the same as 3,3 so those are not two different outcomes.

123456 11,1 = 21,2 = 31,3 = 41,4 = 51,5 = 61,6 = 7 22,1 = 32,2 = 42,3 = 52,4 = 62,5 = 72,6 = 8 33,1 = 43,2 = 53,3 = 63,4 = 73,5 = 83,6 = 9 44,1 = 54,2 = 64,3 = 74,4 = 84,5 = 94,6 = 10 55,1 = 65,2 = 75,3 = 85,4 = 95,5 = 105,6 = 11 66,1 = 76,2 = 86,3 = 96,4 = 106,5 = 116,6 = 12

 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.  ie: Think of flipping a coin.  If you flipped the coin twice, would you expect Heads 50% of the time (would you be surprised if you got two Heads)?  If you rolled the dice 5000 times, would you expect Heads approximately 50% of the time (would you be surprised if you got 5000 Heads)?  This is an example of the Law of Large Numbers.  You are really comparing theoretical probability to experimental probability.

 A sample space together with an assignment of probabilities.  Each will have probability between 0-1.  All together they will sum to 1.  Homework: p296 P1-P9, E1,2,3,7,9,13