Law of Averages (Dr. Monticino)

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Presentation transcript:

Law of Averages (Dr. Monticino) Law of Averages (Dr. Monticino)

Assignment Sheet Read Chapter 16 Assignment #9 (Due April 6th) Read Chapter 16 Assignment #9 (Due April 6th) Chapter 16 Exercise Set A: 1,2,3,4,5, 6,8 Exercise Set B: 1-7 Exercise C: 1,2 Review Exercises: 1-8,10

Overview Law of Averages Sum of Draws Box Models Law of Averages Absolute and relative error Sum of Draws Box Models Roulette example Texas lottery

Law of Averages The law of averages says that if a chance process is repeated a large number of times , then the percentage of times that a particular event occurs is likely to be close to the probability of that event Provided repetitions of the processes are essentially identical and independent of one another

Law of Averages If the experiment is repeated N times and the probability on each repetition that an event A occurs is p, then (#times A occurs)/N gets close to p as N gets large Note, that this not the same as saying (#times A occurs) gets close to being exactly p*N

Examples A coin is tossed repeatedly. You win $100 if exactly half the tosses are heads. Which is better: 2 tosses or 10 tosses 10 tosses or 100 tosses Now you win $100 if the percentage of heads is between 40% and 60%. Which is better:

Examples Suppose you play roulette in Nevada. Which is a better: Suppose you play roulette in Nevada. Which is a better: Spin 50 times and win $1000 if get 40 or more reds Spin 100 times and win $1000 if get 80 or more reds

Sum of Draws For a random process producing real number values, we are often interested in the sum of the numbers produced For example, if gambling, then your net winnings/losses is an important quantity

Box Models A box model is a useful way to represent a complicated chance process Address the following when constructing a box model Which numbers go into the box How many of each number How many draws are being made from the box

Examples Construct a box model for playing “red-or-black” 10 times at a Nevada roulette table, making $1 stakes Construct a box model for betting on “17” twenty times at a Nevada roulette table, making $1 stakes Construct a box model for buying a ticket in the Texas lottery (Dr. Monticino)