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1. Tossing Coins 2. Routes 3. Balls Drawn From an Urn 1

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An experiment consists of tossing a coin 10 times and observing the sequence of heads and tails. a. How many different outcomes are possible? b. How many different outcomes have exactly two heads? c. How many different outcomes have at most two heads? d. How many different outcomes have at least two heads? 2

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A possible outcome is H T H T T T H T H T where H is heads and T is tails. Each coin has two possible outcomes. By the generalized multiplication principle, the total number of possible outcomes is 2 2 2 2 2 2 2 2 2 2 = 2 10 = 1024. 3

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A possible outcome with 2 heads is H T H T T T T T T T. The 2 heads must be placed in 2 of the 10 possible positions. The number of outcomes with 2 heads is 4

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At most 2 heads means there can be 0 heads or 1 head or 2 heads. There is only 1 possible outcome with no heads and that is if all 10 coins are tails. There are C (10,1) = 10 possible outcomes with 1 head. There are C (10,2) = 45 possible outcomes with 2 heads. Therefore, there are 1 + 10 + 45 = 56 possible outcomes with at most two heads. 5

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At least 2 heads means there can not be 0 heads or 1 head. There are 1 + 10 = 11 possible outcomes with 0 or 1 head. There are 1024 possible outcomes total. So, there are 1024 - 11 = 1013 possible outcomes with at least 2 heads. 6

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A tourist in a city wants to walk from point A to point B shown in the maps below. What is the total number of routes (with no backtracking) from A to B ? 7

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If S is walking a block south and E is walking a block east, the two possible routes shown in the maps could be designated as SSEEESE and ESESEES. 8

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All routes can be designated as a string of 7 letters, 3 of which will be S and 4 E. Selecting a route is the same as selecting where in the string the 3 S’s will be placed. Therefore the total number of possible routes is 9

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An urn contains 25 numbered balls, of which 15 are red and 10 are white. A sample of 5 balls is to be selected. a. How many different samples are possible? b. How many different samples contain all red balls? c. How many samples contain 3 red balls and 2 white balls? 10

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A sample is just an unordered selection of 5 balls out of 25. 11

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To form a sample of all red balls we must select 5 balls from the 15 red ones. 12

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To form the sample with 3 red balls and 2 white balls, we must Operation 1: select 3 red balls from 15 red balls, Operation 2: select 2 white balls from 10 white balls. Using the multiplication principle, this gives 13

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