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Understanding Counting Problems: Coin Tosses, Routes, and Urn Draws

This guide explores various counting problems, including outcomes from tossing coins, routes in navigation, and drawing balls from an urn. It provides examples that detail how to calculate possible outcomes when tossing a coin 10 times, analyze routes between points A and B without backtracking, and determine combinations when selecting from an urn containing red and white balls. Each problem includes methods for counting exact, at most, and at least outcomes, providing a comprehensive understanding of basic probability and combinatorial principles.

elizabeth-mcguire
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Understanding Counting Problems: Coin Tosses, Routes, and Urn Draws

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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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