WonLost 1234 Year Number of Games Warm-Up 1) In which year(s) did the team lose more games than they won? 2) In which year did the team play the most games? 3) In which year did the team play ten games?

Math I UNIT QUESTION: How do you use probability to make plans and predict for the future? Standard: MM1D1-3 Today’s Question: What is a permutation and how do we use it to solve statistic problems? Standard: MM1D1.b.

Probability Math I

Let’s work on some definitions Experiment- is a situation involving chance that leads to results called outcomes. An outcome is the result of a single trial of an experiment An event is one or more outcomes of an experiment. Probability is the measure of how likely an event is.

Probability of an event  The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes.  P(A)= The # of ways an event can occur Total number of possible outcomes Total number of possible outcomes

If P = 0, then the event _______ occur. Probability If P = 1, then the event _____ occur. It is ________ It is ______ So probability is always a number between ____ and ____. impossible cannot certain must 1 0

All of the probabilities must add up to 100% or 1.0 in decimal form. Complements Example: Classroom P (picking a boy) = 0.60 P (picking a girl) = ____ 0.40

A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. Experiment: A marble chosen at random.  Possible outcomes: choosing a red, blue, green or yellow marble.  Probabilities: P(red) = # of ways to choose red = 6 = 3 P(red) = # of ways to choose red = 6 = 3 total number of marbles total number of marbles P(green)= 5/22, P(blue)= ?, P(yellow)= ?

There are 3 ways to roll an odd number: 1, 3, 5. You roll a six-sided die whose sides are numbered from 1 through 6. What is the probability of rolling an ODD number? Ex. P 1 2 = 3 6 =

Tree Diagrams Tree diagrams allow us to see all possible outcomes of an event and calculate their probabilities. This tree diagram shows the probabilities of results of flipping three coins.

Use an appropriate method to find the number of outcomes in each of the following situations: 1. Your school cafeteria offers chicken or tuna sandwiches; chips or fruit; and milk, apple juice, or orange juice. If you purchase one sandwich, one side item and one drink, how many different lunches can you choose? Sandwich(2)Side Item(2) Drink(3) Outcomes chicken tuna There are 12 possible lunches. chips fruit chips fruit apple juice orange juice milk chicken, chips, apple chicken, chips, orange chicken, chips, milk chicken, fruit, apple chicken, fruit, orange chicken, fruit, milk tuna, chips, apple tuna, chips, orange tuna, chips, milk tuna, fruit, apple tuna, fruit, orange tuna, fruit, milk

Multiplication Counting Principle ( aka Fundamental Counting Principle ) At a sporting goods store, skateboards are available in 8 different deck designs. Each deck design is available with 4 different wheel assemblies. How many skateboard choices does the store offer? 32

Multiplication Counting Principle A father takes his son Tanner to Wendy’s for lunch. He tells Tanner he can get the 5 piece nuggets, a spicy chicken sandwich, or a single for the main entrée. For sides: he can get fries, a side salad, potato, or chili. And for drinks: he can get milk, coke, sprite, or the orange drink. How many options for meals does Tanner have? 48

Many mp3 players can vary the order in which songs are played. Your mp3 currently only contains 8 songs (if you’re a loser). Find the number of orders in which the songs can be played. 1st Song 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th Outcomes There are 40,320 possible song orders. In this situation it makes more sense to use the Fundamental Counting Principle. 8 The solution in this example involves the product of all the integers from n to one (n is representing the starting value). The product of all positive integers less than or equal to a number is a factorial = 40,320

Factorial EXAMPLE with Songs ‘eight factorial’ The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial. factorial. 8! = = 40,320

Factorial Simplify each expression. a.4! b.6! c. For the 8th grade field events there are five teams: Red, Orange, Blue, Green, and Yellow. Each team chooses a runner for lanes one through 5. Find the number of ways to arrange the runners = = 720 = 5! = = 120

5. The student council of 15 members must choose a president, a vice president, a secretary, and a treasurer. President Vice Secretary Treasurer Outcomes There are 32,760 permutations for choosing the class officers. In this situation it makes more sense to use the Fundamental Counting Principle = 32,760

Let’s say the student council members’ names were: Hunter, Bethany, Justin, Madison, Kelsey, Mimi, Taylor, Grace, Maighan, Tori, Alex, Paul, Whitney, Randi, and Dalton. If Hunter, Maighan, Whitney, and Alex are elected, would the order in which they are chosen matter? President Vice President Secretary Treasurer Although the same individual students are listed in each example above, the listings are not the same. Each listing indicates a different student holding each office. Therefore we must conclude that the order in which they are chosen matters. Is HunterMaighan Whitney Alex the same as… Whitney Hunter Alex Maighan?

Permutation When deciding who goes 1 st, 2 nd, etc., order is important. *Note if n = r then n P r = n! A permutation is an arrangement or listing of objects in a specific order. The order of the arrangement is very important!! The notation for a permutation: n P r = n is the total number of objects r is the number of objects selected (wanted)

Permutation Notation

Permutations Simplify each expression. a. 12 P 2 b. 10 P 4 c. At a school science fair, ribbons are given for first, second, third, and fourth place, There are 20 exhibits in the fair. How many different arrangements of four winning exhibits are possible? = = 5,040 = 20 P 4 = = 116,280

Homework Page 340 #1-4, Page 344 #7-16, 25, 26