Introduction to Counting

Why a lesson on counting? I’ve been doing that since I was a young child!

Introduction to Counting Good question! Actually we’ll be looking at situations more complex than simply counting 1, 2, 3, …

Introduction to Counting Counting in our context refers to figuring out in how many ways we can group or arrange things. Click here to see a scenario where we could apply counting.here (You might wish to right-click on the link and open the file in a new window.)

Counting Example A school club is putting on a festival to raise money. The club is considering several games which would involve wagering money, and the treasurer needs to determine which games would be likely to generate revenue for the club.

Counting Example Game 1 A player pays $1 and flips three coins. If the coins all match (either all heads or all tails), the player wins $3.

Counting Example Game 2 A player pays $1 and rolls a pair of dice. The player wins $5 if the dice come up doubles (both dice are the same number).

Counting Example Game 3 A player places $1 on each of two numbers on a board that has the numbers 1-9. The player then picks two tokens out of a bag which contains nine tokens, also numbered 1-9. The player wins $1 for matching one number, and $10 for matching both numbers.

Counting Example Which game do you think would be best for the club? Which game would you prefer to play?

Counting Example In order to select which game to offer, the treasurer might want to consider the probability of a customer winning each game.

Counting Example Calculating the probability of winning requires determining the possible number of outcomes for each game. We need to know the ratio Number of Winning Outcomes Number of Possible Outcomes

Counting Example One way to calculate the possible number of outcomes is by constructing a tree diagram.

Counting Example Game 1: Flipping Three Coins Each coin has two possible outcomes: Heads or Tails

Counting Example Game 1: Flipping Three Coins Each coin has two possible outcomes: Heads or Tails 1 st Coin: HT

Counting Example Game 1: Flipping Three Coins Each coin has two possible outcomes: Heads or Tails 1 st Coin: HT 2 nd Coin: HHTT

Counting Example Game 1: Flipping Three Coins Each coin has two possible outcomes: Heads or Tails 1 st Coin: HT 2 nd Coin: HHTT THTHTHTH3 rd Coin:

Counting Example Game 1: Flipping Three Coins Follow the branches downward to see the possible outcomes. 1 st Coin: HT 2 nd Coin: HHTT THTHTHTH3 rd Coin:

Counting Example Game 1: Flipping Three Coins For instance, a player could flip Heads and two Tails. 1 st Coin:HT 2 nd Coin: HHTT THTHTHTH3 rd Coin:

Counting Example Game 1: Flipping Three Coins To determine the total possible number of outcomes, look at the bottom row of the tree. There are 8 possible outcomes. 1 st Coin:HT 2 nd Coin: HHTT THTHTHTH3 rd Coin:

Counting Example Game 1: Flipping Three Coins Only 2 outcomes give us matching coins: all Heads, or all Tails. 1 st Coin:HT 2 nd Coin: HHTT THTHTHTH3 rd Coin:

Counting Example Game 1: Flipping Three Coins The ratio of winning outcomes to total outcomes is then 2/8, or 1/4. 1 st Coin:HT 2 nd Coin: HHTT THTHTHTH3 rd Coin:

Counting Example Game 1: Flipping Three Coins In other words, we would expect a player to win ¼ of the time. 1 st Coin:HT 2 nd Coin: HHTT THTHTHTH3 rd Coin:

Counting Example Game 2: Rolling Two Dice Making a tree diagram was relatively simple for the first game, because it only involved a total of eight possible outcomes.

Counting Example Game 2: Rolling Two Dice Rolling dice, however, involves six possible outcomes per die, meaning six branches per die.

Counting Example Game 2: Rolling Two Dice Rolling dice, however, involves six possible outcomes per die, meaning six branches per die This only represents the first die. To represent the second die, we would need to connect six branches to each of the current six branches.

Counting Example Game 2: Rolling Two Dice Rolling dice, however, involves six possible outcomes per die, meaning six branches per die This quickly becomes impractical, on paper or on a computer.

Counting Example Game 2: Rolling Two Dice Another approach is to use a table, with the columns representing one die, and the rows representing the other die. (To distinguish the two dice, we’ll have one red die and one green die.)

Counting Example Game 2: Rolling Two Dice This table shows the possible outcomes ,11,21,31,41,51,6 22,12,22,32,42,52,6 33,13,23,33,43,53,6 44,14,24,34,44,54,6 55,15,25,35,45,55,6 66,16,26,36,46,56,6 Green Die RedRed

Counting Example Game 2: Rolling Two Dice There are 36 possible outcomes ,11,21,31,41,51,6 22,12,22,32,42,52,6 33,13,23,33,43,53,6 44,14,24,34,44,54,6 55,15,25,35,45,55,6 66,16,26,36,46,56,6 Green Die RedRed

Counting Example Game 2: Rolling Two Dice There are 6 winning outcomes ,11,21,31,41,51,6 22,12,22,32,42,52,6 33,13,23,33,43,53,6 44,14,24,34,44,54,6 55,15,25,35,45,55,6 66,16,26,36,46,56,6 Green Die RedRed

Counting Example Game 2: Rolling Two Dice The ratio of winning outcomes to possible outcomes is 6/36, or 1/6. We expect a player to win 1/6 of the time ,11,21,31,41,51,6 22,12,22,32,42,52,6 33,13,23,33,43,53,6 44,14,24,34,44,54,6 55,15,25,35,45,55,6 66,16,26,36,46,56,6 Green Die RedRed

Counting Example Game 3: Picking Two Numbers, 1-9 We could also use a table for this example ,11,21,31,41,51,61,71,81,9 22,12,22,32,42,52,62,72,82,9 33,13,23,33,43,53,63,73,83,9 44,14,24,34,44,54,64,74,84,9 55,15,25,35,45,55,65,75,85,9 66,16,26,36,46,56,66,76,86,9 77,17,27,37,47,57,67,77,87,9 88,18,28,38,48,58,68,78,88,9 99,19,29,39,49,59,69,79,89,9

Counting Example Game 3: Picking Two Numbers, 1-9 At first, it may appear there are 81 possible outcomes. But notice that this includes some impossible outcomes. I.e., we cannot end up with two of the same number ,11,21,31,41,51,61,71,81,9 22,12,22,32,42,52,62,72,82,9 33,13,23,33,43,53,63,73,83,9 44,14,24,34,44,54,64,74,84,9 55,15,25,35,45,55,65,75,85,9 66,16,26,36,46,56,66,76,86,9 77,17,27,37,47,57,67,77,87,9 88,18,28,38,48,58,68,78,88,9 99,19,29,39,49,59,69,79,89,9

Counting Example Game 3: Picking Two Numbers, 1-9 We have also included some duplications. For example, drawing 2 and then 3 is the same as drawing 3 and then 2; we’re only interested in the pairing, not the order ,21,31,41,51,61,71,81,9 22,1-2,32,42,52,62,72,82,9 33,13,2-3,43,53,63,73,83,9 44,14,24,3-4,54,64,74,84,9 55,15,25,35,4-5,65,75,85,9 66,16,26,36,46,5-6,76,86,9 77,17,27,37,47,57,6-7,87,9 88,18,28,38,48,58,68,7-8,9 99,19,29,39,49,59,69,79,8-

Counting Example Game 3: Picking Two Numbers, 1-9 In reality, then, we have 36 possible outcomes ,21,31,41,51,61,71,81,9 22,32,42,52,62,72,82,9 33,43,53,63,73,83,9 44,54,64,74,84,9 55,65,75,85,9 66,76,86,9 77,87,9 88,9 9

Counting Example Game 3: Picking Two Numbers, 1-9 Say a player chooses 1 and 2. We need to know the number of pairs that contain either 1 or 2, but not both ,21,31,41,51,61,71,81,9 22,32,42,52,62,72,82,9 33,43,53,63,73,83,9 44,54,64,74,84,9 55,65,75,85,9 66,76,86,9 77,87,9 88,9 9

Counting Example Game 3: Picking Two Numbers, 1-9 There are 14 winning combinations for matching 1 number. We expect that 14/36, or 7/18 of the time, a player will win $ ,21,31,41,51,61,71,81,9 22,32,42,52,62,72,82,9 33,43,53,63,73,83,9 44,54,64,74,84,9 55,65,75,85,9 66,76,86,9 77,87,9 88,9 9

Counting Example Game 3: Picking Two Numbers, 1-9 There is only 1 pairing that will match the player’s pairing, so we expect that 1/36 of the time the player will win $ ,21,31,41,51,61,71,81,9 22,32,42,52,62,72,82,9 33,43,53,63,73,83,9 44,54,64,74,84,9 55,65,75,85,9 66,76,86,9 77,87,9 88,9 9

Counting Techniques So far, we have looked at two different techniques for counting possibilities: tree diagrams and tables. For more complicated problems, however, we have another powerful tool: The Multiplication Principle

People have stated the Multiplication Principle in a number of ways. Here’s one way to say it: Consider the number of options available every time you make a choice or conduct a trial. Multiply those numbers together to determine the total possible number of outcomes.

The Multiplication Principle Game 1: Flipping Three Coins Each coin has a possibility of two outcomes: Heads or Tails. Because we are flipping three coins, multiply three 2s together. 2*2*2 = 8 possible outcomes

The Multiplication Principle Game 3: Picking Two Numbers This is a more common application of the multiplication principle, where we have no repetition—we cannot have the same number twice.

The Multiplication Principle Game 3: Picking Two Numbers For picking the first number, we have 9 options: the numbers 1-9. For picking the second number, we have 8 options: the numbers 1-9, except the number we picked first.

The Multiplication Principle Game 3: Picking Two Numbers 9*8 = 72 possible outcomes Recall, though, that we have duplications. Every pair of numbers has been counted twice. For example, 1,3 is the same as 3,1. As a result, we need to divide by 2 to get the true number of possible outcomes.

The Multiplication Principle Game 3: Picking Two Numbers (9*8) / 2 = 36 possible outcomes

The Multiplication Principle When using the multiplication principle, and when doing counting problems in general, it is vital to determine whether order matters. If order does not matter, then the multiplication principle will usually give an answer that overcounts, and you will need to modify it. We’ll see more examples of that later in the unit.

Sample Problems 1.One club member has suggested changing the first game by having players flip 4 coins. How many outcomes are possible for this version of the game? (The coins are unique, or we are keeping track of the order in which they are flipped.)

Sample Problems 1.One club member has suggested changing the first game by having players flip 4 coins. How many outcomes are possible for this version of the game? (The coins are unique, or we are keeping track of the order in which they are flipped.) Each coin has 2 possible outcomes (Heads or Tails) so the total possible number of outcomes is 2*2*2*2 = 16 possible outcomes

Sample Problems 2. One club member wants to make the third game more challenging by using 12 numbers instead of only 9. How many outcomes are possible when selecting 2 numbers out of 12?

Sample Problems 2. One club member wants to make the third game more challenging by using 12 numbers instead of only 9. How many outcomes are possible when selecting 2 numbers out of 12? There are 12 options for the first number, and then 11 options for the second number. The order does not matter, though, so we are counting each pair twice and must divide by 2: (12*11) / 2 = 66 possible outcomes

Sample Problems 3. Click here to look again at the Whopper problem. How many different ways can one order a Whopper?here (You might wish to right-click on the link and open the file in a new window.)

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