#### Transcript Probability Intro

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? Number of Games Warm-Up 10 8 6 4 2 0 Won 1 2 Lost 3 Year 4 Math I Day 48 (10-19-09) UNIT QUESTION: How do you use probability to make plans and predict for the future? Standard: MM1D1-3 Today’s Question: How can I find the different ways to arrange the letters in the word “PENCIL”? Standard: MM1D1.b. Probability Math I October 19, 2009 Statistics Box Plots Control Charts Let’s work on some definitions Experiment- is a situation involving chance that leads to results called outcomes. In the previous problem, the experiment is spinning the spinner. An outcome is the result of a single trial of an experiment The possible outcomes are landing on yellow, blue, red or green One event of this experiment is landing on blue. The probability of landing on blue is one fourth. 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 number of ways an event can occur Total number of possible outcomes P(blue)= number of ways to land on blue total number of colors Probability cannot occur. If P = 0, then the event _______ impossible It is ________ must occur. If P = 1, then the event _____ certain It is ______ So probability is always a number 1 0 between ____ and ____. Complements All of the probabilities must add up to 100% or 1.0 in decimal form. 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) = number of ways to choose red = 6 = 3 total number of marbles 22 11 P(green)= 5/22, P(blue)= ?, P(yellow)= ? Ex. You roll a six-sided die whose sides are numbered from 1 through 6. What is the probability of rolling an ODD number? There are 3 ways to roll an odd number: 1, 3, 5. 3 P = 6 1 = 2 Theoretical or experimental? We can calculate what our probabilities should be (theoretical values), but that is not always what happens in a real experiment. We could spin the spinner and land on the blue sector every time (experimental values). That’s not very likely, but it could happen Favorable outcomes Suppose you have the four color spinner-(red, blue, green and yellow. The probability of spinning a red is ¼, but how many reds should you get if you spin it 20 times? 20 * ¼ = 5 times , you should theoretically land on red 5 times in 20 spins. Does that always happen with the spinnerswhy don’t the values always match what you expect? 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. • Calculate P (heads), P(2heads,1 tail), P(tails) Probability: Permutations 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? There are 12 possible lunches. Sandwich(2) Side Item(2) chips chicken fruit chips tuna fruit Drink(3) Outcomes apple juice orange juice milk apple juice orange juice milk chicken, chips, apple chicken, chips, orange chicken, chips, milk chicken, fruit, apple chicken, fruit, orange chicken, fruit, milk apple juice orange juice milk apple juice orange juice milk tuna, chips, apple tuna, chips, orange tuna, chips, milk tuna, fruit, apple tuna, fruit, orange tuna, fruit, milk Multiplication 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, Marcus, to Wendy’s for lunch. He tells Marcus he can get a 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 a frosty, coke, sprite, or an orange drink. How many options for meals does Marcus have? 48 Many mp3 players can vary the order in which songs are played. Your mp3 currently only contains 8 songs. Find the number of orders in which the songs can be played. There are 40,320 possible song orders. 1st Song 2nd 3rd 4th 5th 6th 7th 8th Outcomes In this situation it makes more sense to use the Fundamental Counting Principle. 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320 The solutions in examples 3 and 4 involve the product of all the integers from n to one. The product of all positive integers less than or equal to a number is a factorial. Factorial The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial. factorial. EXAMPLE with Songs ‘eight factorial’ 8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320 Factorial Simplify each expression. a. 4! 4 • 3 • 2 • 1 = 24 b. 6! 6 • 5 • 4 • 3 • 2 • 1 = 720 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. = 5! = 5 • 4 • 3 • 2 • 1 = 120 5. The student council of 15 members must choose a president, a vice president, a secretary, and a treasurer. There are 32,760 permutations for choosing the class officers. President(15) Vice(14) Secretary (13) Treasurer(12) Outcomes In this situation it makes more sense to use the Fundamental Counting Principle. 15 • 14 • 13 • 12 = 32,760 Let’s say the student council members’ names were: John, Miranda, Michael, Kim, Pam, Jane, George, Michelle, Sandra, Lisa, Patrick, Randy, Nicole, Jennifer, and Paul. If Michael, Kim, Jane, and George are elected, would the order in which they are chosen matter? President Is Michael Vice President Kim Secretary Jane Treasurer George the same as… Jane Michael George Kim ? 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. Permutation Notation Permutation When deciding who goes 1st, 2nd, etc., order is important. A permutation is an arrangement or listing of objects in a specific order. The order of the arrangement is very important!! P n! r = (n r )! The notation for a permutation: n n is the total number of objects r is the number of objects selected (wanted) *Note if n = r then nPr = n! Permutations Simplify each expression. a. 12P2 12 • 11 = 132 b. 10P4 10 • 9 • 8 • 7 = 5,040 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? = 20P4 = 20 • 19 • 18 • 17 = 116,280 Classwork Practice Workbook Lesson 6.2 - #12-20 Homework Page 344 #1-6, 25-28