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

 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

 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 =

 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

 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 = 56 possible outcomes with at most two heads. 5

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

 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

 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

 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

 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

 A sample is just an unordered selection of 5 balls out of

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

 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