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5.6 Further Counting Problems

5.6 Further Counting Problems. Tossing Coins Routes Balls Drawn From an Urn. Example Tossing Coins. An experiment consists of tossing a coin 10 times and observing the sequence of heads and tails. a . How many different outcomes are possible?

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5.6 Further Counting Problems

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  1. 5.6 Further Counting Problems • Tossing Coins • Routes • Balls Drawn From an Urn

  2. Example Tossing Coins • 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?

  3. Example Tossing Coins (a) • 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 = 210 = 1024.

  4. Example Tossing Coins (b) • 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

  5. Example Tossing Coins (c) • 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.

  6. Example Tossing Coins (d) • 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.

  7. Example Routes • 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?

  8. Example Routes (2) • 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.

  9. Example Routes (3) • 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

  10. Example Balls Drawn From an Urn • 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?

  11. Example Balls Drawn From an Urn (a) • A sample is just an unordered selection of 5 balls out of 25.

  12. Example Balls Drawn From an Urn (b) • To form a sample of all red balls we must select 5 balls from the 15 red ones.

  13. Example Balls Drawn From an Urn (c) • 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

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