1. Math I 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.

2. Started out with the small diagnostic Then showed the video “What is Probability” from the web site: www.learner.org

3. How many ways can four airlines fly two sizes of jets to three cities?

4. Multiplication Counting Rule (Make a Tree Diagram) 4 times 2 times 3 = 24

5. There are 20 candidates for three different executive positions. How many ways are there of filling the positions? How many options do you have on the first pick? How many options do you have for the second pick? (Answer: 20) (Answer: 19)

6. (Answer: 20 times 19 times 18 equals 6,840) How many options do you have for the third pick? What is the total number of options? (Answer: 18)

7. How many unique 5 digit security codes can be made from the numbers 0 – 9 if we allow the numbers to be repeated? (Answer: 10 * 10 * 10 * 10 * 10 = 10^5 = 100,000)

8. How many unique 5 digit security codes can be made from the numbers 0 – 9 if we do not allow the numbers to be repeated? (Answer: 10 * 9 * 8 * 7 * 6 = 30,240)

9. How many ways are there to flip a coin 10 times? They could all be heads You could have the first 9 tosses be heads, and the last toss be a tail. Etc. How many sides of the coin do we have? How many tosses do we have? How many ways? Answer: Draw the Tree Diagram: 2 *2*2*2*2*2*2*2*2*2 = 2^10 = 1,024

10. Suppose there are five dangerous military missions, each requiring one soldier. In how many different ways can five soldiers from a squadron of 100 be assigned to these five missions? (Answer: 100 * 99 * 98 * 97 * 96 = 9,034,502,400)

11. Multiplication Counting Principle If one event can occur in m ways and a second event can occur in n ways, the number of ways the two event can occur in sequence is m times n.

12. 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

13. 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

14. Class Work Tree Diagram handout

15. A probability experiment is an action through which specific results (counts, measurements, or responses) are obtained. The result of a single trial in a probability experiment is an outcome. The set of all possible outcomes of a probability experiment is the sample space. An event consists of one or more outcomes and is a subset of the sample space.

16. Classical (theoretical) Probability Classical (or theoretical) probability is used when each outcome in a sample space is equally likely to occur. The classical probability of a event E is given by:

17. Empirical (or statistical) Probability Empirical (or statistical) probability is based on observations obtained from probability experiments. The empirical probability of an event E is the relative frequency of event E.

18. 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

19. Law of Large Numbers As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event.

20. Complement of event E The complement of event E is the set of all outcomes in a sample space that are not included in event E. The complement of event E is denoted by E’, and is read as “E prime”. The probability of an event plus its complement has to equal 1.

21. 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 1.00

22. Experiment 1 Break class into groups of 2. Have one person spin penny, and the other keep a record if the penny came up heads or tails. Repeat 7 times. Calculate the probability of heads, P(h), per group. Calculate the probability of heads, P(h), per the class. These calculations are experimental probability.

23. Experiment 1 What is the theoretical probability of “spinning a head”, P(h)? What is the sample space? What is the complement of “spinning a head”? What is the theoretical probability of the complement? What is the theoretical probability of the P(h) plus the complement? Was the whole class experimental P(h) closer to the theoretical probability or individual groups?

24. Experiment 2 Break class into groups of 2. Have one person roll two die, and the other keep a record if the sum of the die. Repeat 10 times. Calculate the probability the sum was less than 5, P(sum < 5), per group. Calculate the probability the sum was less than 5, P(sum < 5), per the class. What kind of probability is this?

25. Experiment 2 What is the theoretical probability of the sum being less than 5, P(sum < 5)? What is the sample space? What is the complement of the sum was less than 5, P(sum < 5)? What is the theoretical probability of the complement? What is the theoretical probability of the P(sum < 5) plus the complement? Was the whole class experimental P(sum < 5) closer to the theoretical probability or individual groups?

26. Class Work Pg 340, # 1 – 13 all

27. Warm Up Make a tree diagram and list the results of a “number” consisting of one of the numbers 1, 2, 3, and one of the letters a, b, c. List the possible options Answer: (3 * 3) + (3 * 3) = 18

28. Multiplication Counting Rule We have been using the Multiplication Counting Rule Use the multiplication rule when order does not matter Examples: ordering a meal, picking people for positions, etc.

29. Addition Counting Rule Use the Addition Rule for events when order does matter You have to calculate the number of possibilities of each “order”, and then add them together.

30. Addition Counting Rule If the possibilities being counted can be divided into groups with no possibilities in common, then the total number of possibilities is the sum of the numbers of possibilities in each group.

31. Addition Counting Rule Calculate the number of ways to make a model number if it has to have one number and one letter Answer: 10 * 26 + 10 * 26 = 520

32. Addition Counting Rule Calculate the number of ways to make a three symbol model number if it has to have one number and two letters and the letters can not repeat

33. Addition Counting Rule Answer: Number first (10 * 26 * 25) = 6,500 Number second: (26 * 10 * 25) = 6,500 Number third: (26 * 25 * 10) = 6,500 Sum: 6500 + 6500 + 6500 = 19,500

34. Addition Counting Rule Calculate the number of ways to make a two symbol model number. One symbol has to be a number, the other symbol can be a number or letter. Number first: 10 * 36 = 360 Number second: 26 * 10 = 260 Total: 360 + 260 = 620

35. Addition Counting Rule Calculate the number of ways to make a two symbol model number. At most, one symbol has to be a number, the other symbol can be a number or letter. Number first: 10 * 36 = 360 Number second: 26 * 10 = 260 Total: 360 + 260 = 620

36. Addition Counting Rule Calculate the number of ways to make a three symbol (numbers and letters) model number if at least one letter is used Letter first (26 * 36 * 36) = 33,696 Letter second: (10 * 26 * 36) = 9,360 Letter third: (10 * 10 * 26) = 2,600 33,969 + 9,360 + 2,600 = 45,656

37. Class Work: pg 340, # 10 & 11 Pg 341 # 10, and Handout # 4-13 – 4-26