Coins Alt - SIMULATED COIN TOSSING Excel simulates 5,000...

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Unformatted text preview: SIMULATED COIN TOSSING Excel simulates 5,000 sets of three coin tosses each. The number of heads in each set is displayed, and these 5,000 numbers are averaged. Pressing F9 will run a new simulation. The simulation requires two types of human input. (i) We assumed that the probability of getting a head when we toss a coin is 1/2. (ii) We assumed that each toss is independent of the results of all other tosses. Toss 2 T T H H T T T H T H T H T H T H H T H T H T T H H H T T T T T H Number of Average number heads of heads 0 1.4818 1 3 3 1 2 2 1 0 2 1 3 2 3 2 1 1 1 2 1 1 1 1 1 2 2 0 0 2 0 2 1 Mathematics for Business De Release 1.53a, 20 Published and Distribu The Mathematical Associatio 2007 by The Arizona Board The University of Arizona. All 1 T H H H T H H T T H T H H H H T T H T H T H H T H H T T H T H T 3 T T H H H H H T T T H H H H H T T T H T T T T T T T T T H T H T Computation of file requires that the a ins Analysis ToolPak a Analysis ToolPak - VB enabled. 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2 2 0 1 2 2 2 1 2 3 3 2 1 1 0 1 1 2 1 2 2 2 3 0 2 2 1 2 T T H H T T H T T H H H H T T T H T T T T H H H H H T T H T T H T T T T H T T H H H T T T T T T H H T T H T T T H H T T H H T T H T T T H H H T T H T H T T H T T H H T T T T H H H H T H H T T T T T H T T H H T H T H H T T H T T T T T T T H H H H T T T H T H H T T 1 1 2 1 2 2 1 0 1 1 1 2 2 1 1 1 2 2 0 1 2 1 1 2 2 2 1 0 1 1 0 3 1 1 2 0 1 1 2 1 2 2 0 1 H H H T T T H T H T T T H T H H H T H H T H T T H H H H H T H T H H T H T T T H T H H H H T T T T H H H T T T H H H T T H T T T H H T H T T H T T T H T T H T T T H H H H H H T H T H T T T T H H T T H H H T H H T T H H T H H H T H T T H T T H T H H H T T H T T H T 3 1 2 0 0 1 2 2 2 0 0 2 3 2 1 2 3 0 1 2 2 2 1 2 2 1 3 1 1 1 2 0 2 2 1 2 1 1 1 3 1 2 3 1 T H T T T T H H T H H T T H H T H T T H H T T T T H T H H H T H H T H T T H H H T T H T T T T T H T T T H H H T H H H T T T T T T H H T T T T T H H T T H T T T H T H H H H T T H H T T H H T T H H T T H T H H T T T H T H H H H H T T H H H T T H H T H T H T H T H T 1 2 0 0 2 1 1 1 2 3 2 0 2 2 3 1 1 0 0 2 1 2 2 1 1 2 0 1 3 3 1 1 2 1 2 0 2 1 3 2 2 1 2 0 H H T H H H H H H H H T T T T H H T H T T T H H T T H T H T H H H H T T H T T H H T H H T H H T T H T H T T T H H H H T T H T T T T T H H T H T H T T T H H T T T T T T T T T H T H T H T H H T H H H T T H H H T H T H H H H H T T T T H H H T H H T H H H H T T H H T 1 3 1 2 1 3 2 2 2 2 2 1 1 2 2 2 1 2 1 1 1 1 2 3 1 0 2 0 3 1 2 1 3 3 0 1 2 1 1 1 1 1 2 2 T H T T T T T H H T H T H H H T T H H T T T H T H T H H H T H H H T H H T T H T T H T H T H T H H H T H T H T H T T H H T T H T H T H H H H T T H T H H H H H H H T T H T T T T T H H T T H T H H H H H T H H T T H T T T T T H T T T T T T T H T T H H T T H T H H T H 0 3 1 1 1 2 0 3 2 2 2 2 1 2 3 1 0 2 2 0 1 0 2 2 2 1 1 1 2 0 2 3 2 1 3 3 1 0 2 1 1 2 0 2 H H T T H H H T T T T H T T H H T H H H H T H T T H T H T T H H H H T T T H H T T H H T T T T H T T H H T H H T H T H T H T T T H H H T T T T T T H H H T T H T H T T H H H T T H H T H H T H H H T T T H T T H H T T T T H T T H H H T T H T T H T H H T H H T H H H T 2 2 0 2 2 1 3 2 1 1 1 1 2 0 2 2 2 1 1 1 2 2 2 0 1 2 1 1 0 2 2 2 2 1 2 1 1 2 2 1 2 3 2 0 H T H H T T T T H H H H H T T T H T H T H H H T H H T T T T T H T H H T H T H T T H T T T H H H H T T H T T T T H H T T H T H H H T T T H H T H H H H H H H T T T H T T H H H H H H T H T T H T H H T H H T T T T H H T T T T T T H T T T T H T T H T T H H T T T T H T 2 2 2 3 1 0 1 1 2 2 1 2 3 1 0 0 2 1 3 1 2 1 1 0 2 3 0 1 1 1 2 2 1 3 1 0 2 2 1 0 1 2 2 1 T T H T H H T T T H T T T H T H H H T T H T H T H H T H T T T H T H H T T H H T H T T H T H H H H H T T T H H T H H H H H T H H H H H H H H H H H H H H H H H H H T T T H H H H H T H H H H H T T H T H T T T T T H H T T T H H T H H H H T H H H T T T T H T H T T H H 1 1 3 2 3 3 1 0 0 3 1 1 1 2 1 2 2 2 2 1 2 1 3 2 2 3 2 3 2 1 2 3 2 2 2 1 1 2 1 1 2 1 2 3 T H H H H H T H T H T T T T T T H H H H T T T T T T T H H T T H H H T T T H H H T H T T T H H H T H T H H H T H H H T H T T T T 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2 1 1 3 2 1 2 2 1 2 1 1 1 2 2 1 1 1 0 0 H T H H T H H T T T H T H H T H T H H H H T T H H T T T T T T T H H T H H H T H H T T H H T H H T T H H H H T H T H H H T T H T H H H T H T H T T H T H T T H H T T H T T T T H T H H H T H T T H T H T H H H T H T H H T H H H T T T H T T T T H T H H H H T H T H T T 2 1 3 3 0 2 2 1 2 1 2 1 2 3 2 2 1 1 3 2 2 2 2 2 2 0 1 1 0 1 0 1 2 1 2 3 2 2 1 2 1 1 0 2 H H H T T T H T T T T T H H T H H T H H T H T T T H H T T H T T H H H T H H T H T H T H H T T H T H H H T T T H T T H H H H H H T T T T H T T H H T H T H H H H H T T H T H T H H H T H T T T H H T T H H H H H H H H H H H H T H T H H T T H T T H T H T T H T H H T T 3 2 1 2 0 1 2 2 1 0 0 2 2 2 2 3 3 2 3 3 1 2 1 0 2 1 2 2 1 1 2 0 2 3 2 2 2 1 1 2 1 3 0 2 T H H T T H T H H H T T H T H H T H T T H T T H T H H H H H T T T T T T H H H T T H H H H T H H H H H H H H H H T T T T T T H T H H H T T H H H H H H T T T T T H H H T T H H T T H T H H T H T H H H H H T T T T H H T T H T H T H H H T H H T H T T H H T T H H H H T 1 2 2 2 2 2 2 2 3 3 2 2 2 0 1 1 0 2 2 0 2 2 1 2 0 3 3 3 2 3 2 0 1 0 0 1 3 2 2 1 1 3 3 1 H T H H H H H T T H H T H T H H H H T H T H T T H H T H H T T T T T H T T H T H H T T H T H H T H H H H T T T T T H H T T H H T T H H H H H T H T H T H T T T H H T T H H H H T T H H T T H H H T H T H H H H H H T H H H T T T H T T H T H T T H H T T T T T H H H H H 1 2 3 1 2 3 3 2 0 2 1 1 2 2 3 2 2 2 2 2 1 2 1 1 3 2 0 3 1 2 0 1 1 1 1 1 1 1 0 3 3 2 2 2 H H T T H T H H H T T T H H H H T T T T T H T H T T T H H T H H H T T H T H T T T H T T H T T H T T H T T T H H H T H H H H T H H H H T H T T H H T T T T H T T T T T T H H H H H H T H H H T H H T T T H H H H H H T T T H H H H H H T H H H H T T T H T T H H T T T T 3 2 0 2 2 1 2 2 2 0 1 1 3 2 3 3 2 2 0 1 1 3 2 2 2 1 1 2 3 1 2 2 1 1 0 2 0 1 1 1 1 2 1 1 H T T T T T T H H T H T H H H T H T H H T T T H H H H T H T T T H T T T T T H H T T H H H H T T H T T H T H H H H T H T H T T H T H T T T H T H T H T T H H T H H T H H T T H T T H H T T H T T H H H T H H T T T T H T H H H T H T H H H H H H H T H T T T T T T T H T 2 2 1 0 1 1 0 2 2 2 3 1 3 2 2 0 2 0 2 2 1 2 1 1 2 2 2 2 2 2 1 1 3 1 1 1 1 0 2 2 0 0 3 1 T H H T T H H H H T T H H H T H T H T H H T H T H T H H H T H T T T H T T T H H T T H T H H T T T H T T T T T H H H H T T H T H T H H T H T H T T T T H T H T H H T T T H T H H T T H H T T T H H H H T T H H T H T T T H T H H T H H H H H H H T T T T H T T H H H H H 1 2 2 1 0 2 1 2 2 1 1 2 2 3 2 1 1 2 0 2 2 1 3 1 2 1 3 2 2 1 2 2 0 1 1 1 2 0 1 2 2 1 3 2 T H H H T T T T H H T T H T H H H T T H T T T H T T T H H T T H T H T T T T H H H H H T H H T T T H T T T T H T T T T H H H T H T T T T T H T H T T H H H H H T T T H T H T T T H H T H H H H H H T H H H T T T T H H H T H H H H T H H H H H T T H H T H H H T H H T T 2 3 1 2 1 2 1 1 2 1 2 1 2 0 1 2 2 2 1 3 0 1 1 2 1 1 1 3 2 1 2 2 1 3 2 0 1 1 3 1 3 2 1 0 H T H T H H H H T H H H H H H T T H T T H T T H H T T H T H T T H T H H T T H T T T T H H H T H T T H H H H T H T H H T T T H T T H T T H T T H 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T H T H T T H T H T T T T T H T H H H H T T H T T T T H T H T H T T T H H T T T T T H H T H T H T T H H H H T H H T T T T T H T T T H H H H H T H T H H H T T T H H H H H T H H T H H H T H H T H H H T H H H T T T T H H H T T H T T T 1 1 2 3 1 0 0 1 3 2 1 1 1 0 3 3 0 3 1 3 0 1 3 1 3 2 1 1 2 1 2 0 1 1 2 2 1 1 2 1 2 1 1 1 H H T T H H T T T H H T H H H H H H T T T T T T H H T H H H H T T T H T H T H T H T H T T H T T H T T T H T H T H T T T T H H T T H H T T T T H H T T T H H T T T T T T H H T H T H H T H T H H T H T T T T T T H T H H H T T T T T H T T H H T T H T T T H H T H T H T 1 3 1 0 3 1 1 1 1 2 2 0 2 1 1 1 2 2 2 1 1 1 1 0 1 1 1 2 2 2 2 0 1 2 1 0 1 1 2 0 3 1 2 1 H H H T H H T H H T H T T H H T T T H H H H T T T T T H T T H T T H T H T H T T T T H T T H T H T T T T T T H H T H T T T T H T H T H T T T T H T T H H H H T T H T H H H T H H H H T H T T H H H H T H H T H T T H T T T T H T H H T T T T H H T H H T H T H H H T T H 2 3 1 2 1 1 1 2 2 1 2 2 1 2 2 0 0 1 2 1 2 1 2 0 1 1 0 2 0 0 3 2 1 3 1 1 2 1 2 2 2 0 2 2 T T T H H H T T T T T T T T H H T T T H H H H T H H T T H T T H H H T H T T H T T H H H H T H T T T H H H T T H H H H T H H H T T H H T H T H T T H H T H H H H H H H H H H H H T H T T H H H H H T H H T H T T H H T T H T H H T T T H T H H H T T T H T T H T T H H T 1 1 1 1 2 2 2 2 2 0 1 2 1 2 2 1 2 2 1 1 2 2 3 1 2 1 1 1 1 2 2 2 2 2 1 3 1 1 3 1 1 3 3 2 T H T H T H T H T H T T T H H H T H T T T H H T T H T H T T T T H T H H T H H H H H H H T T T H H H H H T T H H H H H H H T H T T H H T T H T H T H T H H T H H H H H H T H T H T H T T H H H T T H T T T T H T T T H T T H H T H H H T H H T T H T T T H T H T H T T T 0 2 0 2 2 3 2 2 0 2 1 1 1 2 3 2 1 1 2 0 0 3 3 0 1 3 1 2 1 2 0 1 3 0 2 2 2 2 3 2 2 2 1 2 T H H T H H H T H H T H T H H H H H H H H T T T H T H T H T T H H H T T T H T T H T T H H T T T T H H T T T T H H T T H H H H H H H T H T H T T H T T T H H H H H H H H T T H H H T T T T T T H T T H H T T H H T H H T T T H T T H T T T H T T T H H H H H H H H T T H 2 1 1 0 1 2 2 1 1 1 1 3 1 1 2 3 2 3 3 2 2 1 1 1 1 2 1 0 2 1 0 1 2 3 2 2 2 3 2 2 2 0 1 3 T T H H T T H H T T T H H T T T T T H H H H H H H T T H T T T H T T T H T H T T T H H T H T T H H H H T T H T T T H T T H H T T H H T T H T H H H T T H T T H T H T H T H T H H H H T T H H T T T H H H T H H H H H T T T H H T T H T T T H T T T H H H T T H T H H H T 2 1 1 2 2 2 2 1 0 2 1 2 1 2 1 1 2 2 1 1 2 3 2 1 2 1 1 2 1 1 0 2 0 1 2 2 1 1 2 0 2 2 3 1 T H H H T T H T H H T H T H H T T H H H T H T T T H H T T T H H T H H T H H H T H T H H T T H H H T T H H H H H T T H H H T T T T T T H T H T T T H T H H T T T T H H T T H H T H T H H H H H T H H T T T H H H T T H H H T H H H T T T T T T T T T T T T T T T T T T T 1 1 3 3 2 1 2 1 3 3 1 2 0 2 3 2 1 1 2 2 1 1 1 2 1 2 1 0 0 1 1 2 1 1 1 0 1 2 2 0 1 1 2 1 T H H T T H H T T H H H H T T T T T H T H T T H H T T T H H T T T H T H T H H H T T T H H T H H H H H H T T T H H H H H T H H H T T H T T H H H H T H T T H H T T T T H T H H T H T T T H T T H T H H H H T H H T H H H T H T H H T H T T H T T T T T T H H H H T H H H 2 1 2 1 2 2 2 2 0 2 2 3 3 1 2 2 0 2 3 2 1 1 1 2 2 1 2 1 2 2 1 0 0 2 1 1 1 2 2 3 0 2 2 2 H T T H H T H T H H H T T T H T H T T T H H T H H H H T H H T H T H T H H T H H T T T T H H H T T H T H T H T T T T H T H T T T H T H T H T H H T T H H H H T T T T T T T H T H H H T T H T T T H T T H T H T T T H H T H H T H H H H H H H H T H H H H H H H T T H H T 3 2 1 1 2 1 1 1 2 2 1 1 0 1 2 0 2 1 1 0 3 2 1 2 3 2 3 2 2 2 2 2 2 3 1 2 2 1 2 1 0 2 1 1 H H T T H T T T T H H H H T H H T T T H H H H T H T H H H T T T T T H T H T H T T H H H T T H T H H T T H H H T H H H H T H H T T H T H T T T H T H H T T T H T H H T H H H H T H T T H T T T T T T T H H H H T T H T T H T H H T T H T H T T H T H T H T H T H T H H T 2 1 1 1 2 1 0 0 1 2 2 2 3 2 3 2 0 2 1 1 2 2 2 2 1 0 2 2 2 1 1 1 0 1 2 1 2 2 1 2 1 3 3 1 H T H H H H H T H H T T T H T T H H H H H H T T H H T T T H T T H H H T H H H H H H H T T H T H T H T H T T H T T H H T T T T H T T T T H H T T T H T T H H H H T T H T H T H H H T H T H T T H H H T H H H H H H T T H H H H H T H H H H H T T T H T H T H H H T T H H 2 1 2 2 2 2 1 2 2 2 1 1 1 3 2 1 2 1 1 3 2 2 1 1 2 3 1 1 1 3 0 0 2 3 2 2 1 2 3 2 2 1 3 2 H H H T H H T T H H T H T H T H T T H H T H H H T T H T H H H T T H T H T H T H H H H T T T T T H T H H T T T H H H H T H T H T H H H H H H H T T H H T T H T H T H T T T T H T H T T H T H H H H T H H T T T H H T T H T T H H T H T H H H H T T H H T T T T T H T T T 2 1 1 1 2 2 2 2 2 1 1 3 1 2 1 2 2 0 2 2 1 2 3 3 1 2 2 1 2 3 3 0 0 3 1 2 0 2 0 1 2 1 2 0 H T H H H T T T T T T H H H T H T H H T H H H H H H T T T H H T H H T T T H T H T T H T H T H T H H T H H H H T H T T T T T H T H T H H T H H T H H H H T H H T H T T T T T H T T H T T H T H H H H H T H T H H H H H H T H T H H H T H T H T T T T T T T H H H T H T T 2 1 2 1 3 1 1 2 2 2 2 1 3 1 1 2 1 2 3 1 2 2 2 3 2 3 1 1 1 3 2 1 1 2 1 0 1 2 1 2 0 1 2 0 T H H H H T H T T T T H T H T H H H H T T T H H T T H T T H T H H H H T H T H H H T T T H T T H T T T H H T T T H H T H T T H H T T T T T H H T H T H T H H H T H T T H H H H H H T H T H H T T H T T T T T T T H H T T H T H H T H T H T T T T H H T T H H H H T T T H 2 1 2 2 2 1 1 1 2 0 0 1 1 2 0 2 2 2 2 1 1 0 2 2 0 2 2 1 1 1 1 1 3 3 2 0 3 1 2 3 2 1 1 2 H H T H T T H T H H H T H H T T H H H H T T H H T H H H T T T H T H H T T H T H H T H H T T T H T T T T H H H H H T H H H T T T H H T T H H T T T T T H H T T T H H H T T T H T T H T T H T T T H H H T H T H H T T T H H T T T T T T T H H H T H T T H T T T T H H T H 1 2 0 2 1 0 1 0 3 3 3 1 3 1 2 2 2 1 1 2 2 1 1 1 1 2 1 1 1 1 1 2 2 1 1 1 1 2 1 1 2 1 2 2 H H H H T H T H H H H H T T T H H H T H T H H H T H T T H T H H H H T T H H H H H T H H T T H T H T H T H T T T H H T H H H T T H H H T H T T T T H T H H H T H H T T H T T H T T T T H H H T T T T H T T H T T T H T H H H H H H H T H H H H T T T H T H H H H H H H T 1 1 2 2 2 2 1 1 2 1 2 1 1 2 0 2 2 3 0 2 2 3 3 2 2 2 0 1 2 2 2 2 2 2 1 1 3 2 2 3 2 1 3 1 H H T H T T T T H T H H T T T T T T H T H T T T T T T H H H H H H H H T H H H T H T T T H H T T T H T H T H H H H T T H H H H T H H H H T H T T T T T T T H T H T H T T T H H T T T T T H T H H T H T H T H T H T H T H H T H T H T T H H H T T H H T H H T T T H T T T 2 2 0 1 1 1 1 2 1 2 2 3 1 1 0 2 1 2 2 1 3 1 2 1 1 1 0 2 2 2 1 1 2 3 1 2 2 2 1 0 2 1 1 0 T T T T T T H T H T T T T H H T T T H T T T H T H T H T T H T H H H H H T T T T H H H T T H T H T T T T H H H T T H H H H T H H T T T T H H H T T H H H T H T T T T T T H H T T H T T T H H T H T H H T H T H T H H H H H H T H T T T T H H H T T T T T H T T H T T H T 1 1 0 1 1 1 1 1 2 2 2 0 1 2 3 1 2 1 3 2 1 1 1 1 2 1 2 0 1 3 2 2 1 2 1 1 1 0 0 1 2 2 2 0 H H H H H H T H T H H H H T T H T T H T T H T H H H T T H T H H T T T H H T T T H T T H H T H H T T T H H H H H T T T H T T T H H H T H T H H H H H T H H H T T T H T H T H T T T H H H T T T H H H T H T T T H H H T T T H H T T T H H H H T T T T T H H H T T T H H T 2 2 3 3 1 1 0 3 2 3 2 3 1 0 0 3 1 1 1 1 1 3 1 2 1 2 2 2 3 2 1 2 1 1 0 2 2 2 0 1 1 2 1 1 H H H T H H T T T T H T T T T T T T T T H T H H H T T H H T H T T T T H H H H T T H T H H T T H H H T H T H H H T H T T T H H T H H T T H T T T H H H H H H T T T H H T H H H T H H T H T H T H T T T H T H T T T T T T T H T H T H T H H H T T T H H H H T H H T T H T 3 2 1 2 2 3 0 2 0 1 2 2 0 2 0 0 0 1 1 0 2 2 1 2 2 1 0 2 3 2 2 1 1 2 1 2 2 2 3 1 1 2 2 1 H T H T H H T H T H T T T H H H H H T H T H T H H T T H H H H T T H T H T T H H H T T T H T H H T H T H T H T T T H T H T T T T H T T T H H H T T H T H H T T T T T H H H H H H H T T T H H H H T H T H T T T T T T T T T H H T H H H T H T T T H H H T H T T H T H H H 3 0 2 1 2 3 1 3 0 3 0 1 0 2 1 2 1 1 0 1 1 2 1 1 3 2 2 1 2 2 1 1 2 2 1 1 1 0 2 3 2 2 2 2 H H T H H T T T T H H H H H H T T T H T H T H H T T H H T H H T H H T T H T T H T T H T T H T H H T T T H H H T H T T H H H H H T H H T T T T T T H T H H T H H H H T H H H T T T T H H H T H H H T H T H H H H T T T T T H T H H H T T T H H H H T T T T H H T H H T H 1 2 1 3 3 0 1 1 2 2 3 1 3 2 2 2 1 1 2 1 1 2 2 2 1 1 1 1 0 3 2 2 3 1 1 1 2 2 1 2 2 2 1 1 T T H H H H T T H H T H H H H T T T H T H T T H T H T T H H H T H H T H T H T H H T H T T H H H T H T T T H T H T T H H H H T T H H T T H T H H H H H H H H H T T H H T T H T T T T H T T T H T H H T H H T H T T H H T H T T H T T H H T T T H T T T T T T H T T T H H 0 1 3 2 1 2 1 0 2 3 0 3 2 1 3 1 1 2 2 0 3 1 0 2 1 1 2 2 2 2 2 2 2 2 1 1 0 2 2 1 1 1 2 1 T T H T T H T T T T T T T T T T T H H T T T H T T H H T H H T T H H H H T H H T T H T T H H H H T H H H H H H T T T T T T H T H H T T H T T T H H H T T H T T T H T H H T T T H T H T T H T T T H H H H T T T T H T T H H H H H T T T T T H H H H T T H H H H H H T H T 1 2 2 1 1 2 1 1 2 2 2 1 0 0 0 0 1 2 1 2 2 1 2 2 0 1 1 1 2 3 1 1 3 1 1 2 2 2 3 2 1 1 1 1 H H T T T H T T T H T H H T T H T T T T T H H T H T T H T H T H T H H T H H T H H T H T H T T T H H T H T T H H T H H H H H T H T H H H T H H H T T T T H T H H T T T H T T T T T T T T T H H T T H H H T H T T T T T H H T H T T H H H 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T T H T H T H T H T H H T T T T T H H H T H H T T H H T T H T H H H H H T H T H T T T T 1 0 1 1 3 1 3 0 2 1 3 2 1 2 0 1 0 1 1 2 0 2 2 1 1 1 2 0 2 1 1 2 2 1 1 2 0 2 2 1 0 1 1 2 H T T H H H H T T H T H H H T T T H H H H H T T T T H T T T H T H H H T H H H H T H T T T H T T T T T H H H T T H H T H H H T H H T H H H T H T T T H T H T T H T T H T H H H T H T T T T T T T H H H H H H H T H T H T T H H T T T T H H H T H H H T H T H T T T T T T 2 1 0 1 1 1 1 1 2 3 1 2 3 3 1 1 2 2 2 2 2 2 2 1 1 0 2 1 1 1 2 1 3 2 1 2 1 2 2 1 1 2 1 0 H H H H H H H H T T H H H H H H T T T T T T T H H H T T H T H T H H H T H H T H H T T T T H H T T H T H T T H H T T H T H H T H T H H T T T T H T T T H T H T H T H T H H H H T H T H T T T T T H T T H T H T H H H H H T H H H H H H T H T T T T H H H T T T H T H T T 2 2 3 1 1 2 1 2 1 0 2 3 1 2 2 2 2 2 1 2 0 2 2 2 2 2 1 1 2 0 1 1 1 3 2 2 1 2 0 3 2 2 1 0 H H H H T T T T T T T H T H T H H T H H T T T T T T H T H H H T T H T H T T T T H H T T H T T H H H T T H T T H T T T H 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T H T H T H H T 3 1 3 2 1 1 1 1 1 1 1 2 3 1 3 1 0 2 1 2 1 2 3 1 2 2 2 2 2 1 2 0 3 2 2 2 0 1 0 2 2 2 3 1 T H H H H H T H H T H T H H T H H H T H H H T T H H H T T H T H T T T T H T H H H T T H H T H H T T T T H T H T H H T T T T H T H T T T H H T H T T T H T T H H T T H H H T H H H H T H H H T T H T H T H H H T T T T H T T T H T T H H H T T T T H T H H H H H H H H H 2 2 2 3 2 2 0 1 3 0 3 0 3 3 1 1 1 1 1 2 2 1 0 1 2 2 2 2 1 1 0 2 0 1 1 2 2 1 3 3 3 1 2 3 T T T H T T H H T H H T H T H H T T T H T T H H T T H H H H H H T H H T T T H T T T T T T H T T T T T T T T H T T H T T T H H H H H H T H T H H H H H T T T T H T T H H T T H T H T H T T H H T H H H H T T H H H H T H T T T T H H H H T T H H T H T H H T T T H T T T 1 1 1 1 0 1 2 1 1 2 3 1 1 1 2 2 1 2 1 3 1 1 2 1 2 1 3 3 2 2 3 2 0 2 1 2 1 0 2 1 1 0 1 0 H T H T H T T T T H T H T T T H H H T T H T T T T T H H T T T H H H H T H H T H H T T H T H H T H H T T H H H T T T T H H T H T T T H H H H H T H T H T H H T H T T T T T T T T H T H H H H T H T T T T T T H H H T H H H T T H H T T T H T T H H T T T H T H T T T H H 2 1 3 1 3 2 0 1 1 2 1 1 0 0 1 3 3 1 2 1 2 0 1 2 2 1 2 1 2 0 1 2 3 2 1 1 2 1 1 1 1 0 1 2 T H H H H H T T H H H H H H H T T T H T T H T T T H T T T H T T H H H H T H H H H H H H H T H H H H H T T T T H H T T T T T T H T T H T T H H T H T T H H T T H T T T H T T T H T H T T T H T T H H T T T H H T T T T H H H H T T T H H T H T T H T H T T T H T T T H T 1 2 2 2 2 3 1 0 2 2 1 2 2 2 2 0 0 0 1 2 1 2 2 0 0 2 2 1 1 2 0 1 3 1 2 2 0 1 2 2 1 1 2 2 T H T T T T T H H H H H T T H H T T T H T T T H T H H H H T T T H T H T T H T T H T T T T T T T H H H T T T T T T T H H H H T H T T T H T H T H H T H T T T T T T H H T T T H T T T T T H H H T T H H T T H T T H T T H T T H T H H H T T T H H T H T T T T T H T T H T 0 1 0 0 2 2 2 1 1 2 2 1 0 1 2 2 2 1 0 3 0 0 1 2 1 3 2 2 2 0 2 1 1 1 1 0 0 2 1 1 1 0 2 0 T T T H T T H H T T T T H H H T T H T T H T H T T T T H H H T T H T T H H H H H H H H T H T T T T H T H H T T T H H H H H T H T T H T T H T T H H T T T H T T H T H H T H H T H H H H H H H T H T H T T H H T H T T H H H T T T T T H T H H T H T T H T T T T T H T T T 2 1 1 2 1 2 1 3 1 1 0 0 3 3 2 2 1 1 2 1 2 1 1 0 1 0 1 2 3 2 0 1 2 0 1 2 1 2 2 1 3 2 1 1 T T T H T T H T H H H T H H H T T T T H H H T T T H H H T H H T H H T H T T T H H T H T H H T H H T H H H H T H H H H T H T T H H T H T T T T H H H T T T H H H T T H T T T H T H T T H H H T H T T H H H H H H H H T T T H H H H H H T T H H H T H H T H H T H T T H T 2 1 0 3 2 1 2 2 2 2 2 2 3 3 3 1 2 1 0 2 2 2 2 1 1 2 2 2 1 3 2 1 1 3 2 2 1 1 1 2 1 0 3 0 H H H H T H H T T T T T H H H H H H H T H H H H H T H T H T T T H T H H H H H H H T H T H T H T T H T H T T T H H H H H H H H T T T T H H T H T H T H T T T T T T H T T T T T H H H T T H H H H H T H T T T T T H T H H T H H T T H T H H H T H H T H T T T T H T H H H 3 2 2 1 1 3 2 2 1 0 1 1 2 2 2 2 3 2 3 1 1 2 2 2 2 1 2 1 3 1 1 1 2 0 2 1 1 2 1 2 1 1 2 2 T T H H T H T H T T T T H T H T H T H H H H T H T H H T H H H H H H T H H T H H T H H T T H T T T T T T H H T H T T H T T H T H H T H T T T T T T T H T H T H T T T H T T H T T T T H T T H T H H H H H T H T T T T T H T H T H T T H H H H T H H T T T T H H H T H T T 0 1 2 1 0 2 0 2 2 2 1 2 1 1 2 0 1 1 1 3 2 2 1 2 0 1 2 1 2 2 2 2 3 1 1 1 1 1 3 2 0 3 1 0 T T H H T T H T T T H H T T H T T T H H H T H H H T T H T H H H H H T H T H T H T T T T H T T H H T H H T T H H T T H H H T T H T T T H H H T T H T H T H H T H H T H H H T T T T T T H T H H H H T H T T H T H H T H H H T H T H H T T H T H H T H T T T T H T T H H T 1 0 1 3 1 1 3 2 1 0 3 2 0 1 2 2 2 0 2 3 2 0 2 2 3 2 0 1 2 1 3 2 2 3 0 2 1 1 2 2 1 1 1 0 H H H H T H H T H T T T T T T H T H T T T H H T T T H T T T H T H T H H T H H T H H T H T H T T T T H H H H H T T T T T T T T T H H T H T H H H T H H H T T H H H H H T H H H H H T H T T H T H H T T T T T H H T T H H T H T H T T H T T H T T H T H T T H T H H T T H 2 2 2 1 0 2 2 2 3 1 1 0 0 0 1 2 0 1 1 1 1 3 1 2 0 1 3 1 0 2 2 1 2 0 3 2 1 3 2 1 3 2 1 3 T H H T T T H T T H T H T H H T T T T H T T T H T H H T T H T H T H H T T T T H T H T H T T T H H H T T T H T H H T T T T T H H H T H H H H H H T H T H T H T H H H T T T T H T T H T T T T H H H H H T T H T T T T T T H T T T T H H T T H H T H H T H H H H H T H H H 0 2 1 1 1 1 2 1 1 3 1 2 1 2 1 0 0 0 1 2 2 0 1 2 1 3 3 1 0 3 1 2 1 3 1 2 2 2 1 2 0 2 2 2 T H H H T T T T H H T H T T H H H T T H H T H T T T H T H T T T H H H H T H H H H H H H T H H H T H H H H H T H H T H H H H H T H H H H H H H T T T T T T T T H T H T T H T H H H T T H H T T H H H T H T T T H H T T T T T H T H H H H T H T H H H H T T T H T H H H T 1 2 2 3 1 1 1 2 3 3 0 3 1 0 2 3 3 1 1 1 2 1 3 1 2 2 3 1 1 1 0 1 2 2 2 2 0 2 2 1 3 2 3 2 T H T H T H T H H H H H H T H T T T H T T T T T T T T T H H T T T T H T T T T H H T T T H T H H T H T T T H T T T T T T T T H T H H H H H T T T T T T H T T H T T T T H T T T T T H T H T H T T T H T H T H H H T H H H H H T T H H T T T T T T H H T T T H T T H H H T 1 2 1 3 0 3 0 1 1 3 1 2 1 1 2 1 0 1 3 1 2 2 1 1 2 1 0 0 1 1 0 1 1 1 2 0 0 1 0 2 2 1 1 0 H T H T T T T T H T T H H H H H H H H H T T H H T H H T T H T H T T H T H T H H H H T T T H H H T H H T H H H T T T T T H H H H H H T H T T H T T H T H T H T T H H T T H H T T H H H T T H H T T H H T H H T H H T H H T H H H T H H T T H T T T T H H H T T T T H T T 2 2 3 1 0 2 2 0 2 2 2 1 2 2 1 2 3 2 3 3 1 2 2 3 0 2 3 0 0 3 0 2 0 1 2 1 3 1 1 1 2 3 0 0 T T H T H H H T H T T H T T T H H T H T T T H H H H T H T T T H T T H T T T H T H T H T T H T H H H T T T H T T T H T H H H H H H T H T H H T H H H T T T T H H T H H T H T H H H H T T T H H T H H T H T H H H T H T H T H H T H H T T H T T H T T H T H T H H H T T T 1 2 1 1 2 3 2 0 2 2 0 2 0 2 1 3 2 2 2 2 1 1 3 1 3 3 0 2 2 1 0 2 0 0 3 1 1 1 3 1 3 0 2 1 H T H H T T H H H T H T T H T T H T T H T H T T H H T H H T T T T T T H T T H H T H H H T H H T T T T T T T H T T H H H H T T T H T T H T T T H H H H T H H H H T T T T T H H T T T T H H H H H T T H H H H H T H H T T H T T H H T T T T T T T H T H H H H T T H T H H 1 1 2 2 1 1 2 2 1 0 3 1 1 3 2 1 3 1 0 1 2 1 0 2 2 1 0 2 2 1 1 0 2 1 2 3 1 1 1 1 1 2 3 2 T H T T H T T H H H T H T T H T T T H T H H T T T T H T T H T T T H H T H T T H H H T T T T H H T T H T T H H H H T T T H H H T H H H T T H H H H H H H T T H T H T T T H T T T T T T T H H T H T H T T H H T H T T H T T H H T T H T H H T H H H T T T H H T T H H H T 0 1 1 1 2 1 1 2 1 3 1 2 2 1 1 1 1 1 3 0 2 3 2 0 0 2 2 2 2 2 2 2 1 1 2 0 3 1 0 1 3 2 1 0 T H H H T H T T H T T T H T H T T T T H H H H T H H T H H H H H H H T H T T T H H T H H T T T T T H H T T T H T H H H H T T H T H H H H H H T H T T H T T T T T T T T H H H T T H H H H T T H T T H H T T H H H T T H T H T H T T T T H H H H H T H H H H H H H T H H H 1 2 2 2 0 2 2 0 1 1 2 0 2 2 3 2 0 0 2 1 3 2 3 1 2 2 0 3 2 2 3 2 1 2 1 2 1 1 1 3 2 2 2 2 H H T H T T H T H T T H H H H T T T T H T H T T T H T T H H H H T H H T T H H T T H T T H T T H H H T H H T T H H H T T T H T H T H H T T H H T T T H T T H H H H T H T T H T H T T H H H H T H T H T H H H H T H T H T H H T T H H T T T T T T T H H T H H T H T T H T 2 1 1 3 2 2 1 2 2 1 0 3 3 3 2 0 1 1 1 2 1 3 1 0 1 3 1 0 1 1 2 1 0 3 3 1 2 2 2 1 0 2 1 1 H T H T T H T T H T T T H T T T T T T T H H T H T H T T T T H T H T H H T H H T T T H T H H T T H T H H H T H T H T H T H H H H T T H H T H H T T T T T T T T T T H H T T T H T H T T T H H T H H H T T H H T H T H T T T H T H T H T T H T T T T T H T H T T H T T H H 3 1 1 0 2 2 1 2 3 1 1 0 3 1 1 1 1 2 1 1 1 2 1 3 0 3 1 0 1 0 1 0 1 0 2 1 1 2 2 1 0 0 3 1 T T T T T T T T T H H T T H T T T H T T H H T H T T T H H H H H T T H H T H T H H H T T H H H T T T T H H H T T T T H H T T H H T H T T T H H H H T T H H T H T H T T T H T H H H H T T T H H T T T H T T H H T T T H H T T H T H H T H T T H H H T T H H T T H T H H H 2 2 1 0 0 1 1 1 1 2 2 0 0 2 2 1 0 1 2 2 1 2 1 1 1 2 1 3 2 1 2 3 2 0 2 2 2 1 0 2 2 2 2 2 T H H T H T T T T H H T T H H T H T T H H T H H H H T H T T T H T H H H H T H H H T T H T H T H H T T H T T H T H H H H T T T T H H T H H H H T T T T H T T H H T T H H H T T H H H H T T H T H H T T T T T H T H H H H H H T T T T T H H T H H H H T T T T H H T H H H 1 3 2 1 2 1 0 2 1 1 2 0 1 2 3 1 2 1 1 2 3 2 1 2 2 2 1 2 1 0 1 3 1 2 2 2 1 0 3 3 2 1 1 3 H H T T H T T T H T T H H T T T T T T H T H H H H T T T H H H T T H T T T H T H H T T H H H H H T H H T T H H T H T H H H H H T T T H T H H H T T T H T H H H H T T T T T T H T H H H T H T H T H H T H T H T T H H H H T H H H T H H H T T T T T T H T T T H H H T H H 3 3 2 1 2 1 2 0 2 2 1 2 2 1 1 1 2 2 2 2 0 2 3 2 2 2 2 1 1 1 2 0 1 2 2 1 0 1 1 2 2 0 2 2 H H T H H T T T T T H H T T H H H T T T H H T H T H T T T T H T T H T T H T T T H H T T T H H H T H T T H T T T T H H T H H H H H T H H H T H H T T T T T H T H T T H H H H T T H H H T H T T H T H H T T H T H H T H T H H T T T T T T H H H T H H T T T T T H H T T T 2 3 2 2 2 1 0 1 1 1 2 1 0 2 2 2 3 1 2 1 3 2 1 2 1 1 1 1 1 1 2 0 1 3 0 1 1 0 1 2 3 2 0 0 H T H H T T H H H T H T H T T H H H T T H T H H T T H T T H H T T T T T H H H H T T T H T H H H T T T H H H H T H T H H H H H H H H H T T T H T H H T T T T T T T T T T T T H H H H H H T T H H T T H T T T T T T T T T T T H T H H T H T T H H H H T T T H T T T H T T 2 2 3 3 0 0 2 3 2 1 3 0 2 0 1 2 2 2 1 1 2 1 3 1 1 1 2 1 1 2 2 1 1 1 0 0 1 2 1 1 0 1 1 2 H T H H H H H T T T T H T T T H T H H T T H T H T T H T T H T T H H T T T T T T H H H H T T T H H H H H H H T T T T H H H H T T H H T T H T T H H H T H T H T T T T H H H T T H T H H H T H H H H T T H H T T H T T T H T H H H H T H H T T T H H T H T T T H H T H T H 1 1 2 3 2 3 3 2 2 1 0 2 1 0 1 3 1 2 1 1 1 3 1 2 2 0 2 2 1 2 0 2 2 2 1 0 0 0 2 2 2 2 1 3 T T T T H H T H H H H H H T T T T T T T H T H T T T T H T H T T H H H H H T H T T H H H T T H T T T T T H H H H H T H H T H T H T H H T H T T H H H H T H T T T H H T H H H H H T T H T H H T H H H T H H H H T T H T T T H H T T H H T T T T T H T H H T H H T T T T H 0 0 2 0 2 2 0 2 3 3 2 3 3 1 2 1 0 2 0 1 1 2 3 0 1 1 1 2 1 2 1 0 3 1 2 2 2 2 2 1 1 2 2 3 H H H H H T H T T H T T T T T H T H T T T T H T T H H H T H T H T T T H H T T T T T H H T T T H T H H H T T T H H T T T T T T T T T H T T H T H T H H H T H H T T H T T T H T H H T T H H H T T T H T T T T T T T H T T H H T H H T T T H H T T H T T H T T T H T T H T 2 1 1 3 2 2 2 1 0 2 0 1 1 0 0 1 0 2 0 0 1 1 2 1 1 2 1 2 1 3 1 2 1 1 1 2 1 1 0 1 0 1 2 2 H H H H H T T H H H H T H H H H H T H T H T H T H H H H T H H T T T T H T T H H T T H H H T T H T T T H T H H T H H H T T H T T T T T H T T T H T H T H T H T T H T H H T H T T T H H H T T T T H T H T H T T H T H T T H H H H H H H T H H H T T T H H H T H T H T H H 2 2 2 3 1 0 0 2 2 2 3 0 3 2 2 2 1 2 1 0 2 1 2 2 2 2 2 2 1 3 2 1 0 1 1 2 2 0 3 2 1 1 2 2 T H T H T H H H T T H H T T T H H H T H T H T T H T H T T H H T T T H H T H T T H T T T T T T H T H T H T T T T T T T H H T T T T T T H H T T T T H T H T T H H T H T H H T H H T T T H T T T T T H T T H T H T H T H H T T H H H H T H T T H H T H H T T T H T T T T H 0 1 0 3 0 2 1 2 0 1 1 1 1 0 1 2 3 1 1 2 0 1 1 2 3 1 1 1 0 2 2 2 0 1 3 2 0 2 1 1 2 0 1 2 T H T H H H H T H H H H T H T H H T T T T T H T H H H H T T H T H H H T T H H T T T H T T T T T H T H T H H H H T H H T T T H H T T T T T H H H T T H H T H T T H H T T T H T H T T H T T T T T H T H H T T T T T T H H H T H T T T T H H T H H T H H H H H H T H T H T 0 1 1 1 2 1 2 0 3 2 3 3 0 2 1 1 1 0 2 2 1 0 2 0 1 2 2 3 1 0 3 2 1 3 2 1 2 3 2 0 1 1 2 1 H H H T T T T H T H T H H T H T H T H T T H T T T H T T H T T H H H T T H T T H T H H H H H T T H T H H H H H T H T H H H H H H H H T H T T H T H T T T T T H T T T T T T T T T H H H T T H H T H H H T H T T T T H T T H T H T H H T T H T T T H T T H T H H H T H H T 3 3 2 0 1 1 2 2 2 3 2 1 3 0 2 1 2 2 2 1 2 2 1 1 1 2 1 0 3 0 0 1 2 1 1 1 1 1 1 2 0 2 2 1 H T H T H T T H T T H H H T T H H T T T T T H H H H H H T H H T T T H T T T H T H T T T H H H H H H T T T H T T T T T H H H T H T H H T H T H T T T H T H H H H T T H T H T H T H T H H T T H H T T H H T H H H T T T T H H T T T H H H T H T T H T H T H T T H H H H H 3 1 3 2 2 1 1 2 0 1 2 2 1 1 1 3 2 1 0 1 1 2 2 1 2 2 3 2 0 2 2 0 2 1 3 1 1 0 2 1 3 1 2 1 H H H H H T H H H T H T T T H T T H H T H H T H T H T H H H H H T H T T H T T H T T H T H H T T H T T T H T H H T H T T T T H H H H T T T H H T H T H H H H H H T T H H T H H H T T H H T H H H H H T T T H H H H H H T H H H T H T H H H T T T T H T H H T T T H T T T 2 2 2 2 2 1 2 2 3 1 2 1 0 2 2 1 1 2 3 1 3 3 1 1 1 2 2 2 3 1 2 2 1 3 1 2 2 0 1 2 1 1 2 1 H T T H H H H H T H H H H H H T H T H H H H H T T H H H T H H H H T H H T T T H T T H T T T T H T H T H H T T T H H H T T H T H H H H T H T T H H H H H T T T T T H H H H T H T T H H T T T H H T T H T H T H T T T T H H H H H T T H H T H T H 1 1 2 2 2 1 2 2 0 3 2 2 2 1 2 1 2 1 3 1 2 2 2 1 1 2 2 2 2 2 2 3 2 1 3 3 0 1 0 2 Mathematics for Business Decisions, Part 1a Release 1.53a, 2007 Published and Distributed by The Mathematical Association of America 2007 by The Arizona Board of Regents for he University of Arizona. 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This note was uploaded on 04/02/2008 for the course MATH 115a taught by Professor Ekstrom during the Spring '08 term at University of Arizona- Tucson.

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