1. Math-2 Lesson 10-1 Probability

5. Definitions Sample Space: the set of all possible outcomes for an experiment. Outcome: A possible result of a probability experiment is called an outcome You may have noticed that for each of the experiments above, the sum of the probabilities of each outcome is 1. This is no coincidence. The sum of the probabilities of the distinct outcomes within a sample space is 1.

6. Diagrams Diagrams are often the key to getting started on a problem. They can clarify relationships that appear complicated when written. A tree diagram is a type of systematic list that is used to organize information spatially.

7. ___ ___ ___ Given the first letter above, the second letter could be either: Arranging 3 Objects in Order How many ways can you arrange the letters A, B, and C in order? A, B, and C in order? A B or C Any one of the 3 letters could be the 1 st letter. The only option for the 3 rd letter in this case is: C ABCABCABCABC BC

8. Given the first letter above, the second letter could be: Arranging 3 Objects in Order How many ways can you arrange the letters A, B, and C ? A, B, and C ? A B C B or C A or C A or B Any one of the following 3 could be the 1 st letter. The only option for the 3 rd letter in each case is: ABC ACB BAC BCA CBA CAB SIX ways

9. “Tree diagram” How many way can you arrange the letters A, B, and C ? A, B, and C ? A B C B C A C B A C B C A A B ABC,ABC,ABC,ABC, ACB,ACB,ACB,ACB, BAC,BAC,BAC,BAC, BCA,BCA,BCA,BCA, CBA,CBA,CBA,CBA, CABCABCABCAB SIX ways We call this a tree diagram. ABCABCABCABC

10. Multiplication Principle Multiplication Principle states: If an event occurs in m ways and another event occurs independently in n ways, then the two events can occur in m × n ways.

11. What is the sample space?

12. Sample Space: All of the experimental outcomes

14. Your Turn 1. Draw a tree diagram representing the outcomes for flipping a coin then tossing a die. a. How many outcomes are there? b. What is the probability of getting heads, then an even number? *this is problem 1 on your homework

15. Using the Multiplication Principle If a license plate has three letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. L L L # # # How many possibilities for the 1 st position (letter)? 26

16. Using the Multiplication Principle L L L # # # How many possibilities for the 2 nd position? 26 * 26

17. Using the Multiplication Principle L L L # # # How many possibilities for the 3 rd position? 26 * 26

18. Using the Multiplication Principle L L L # # # How many possibilities for the 4 th position (number)? 26 * 26 * 10

19. Using the Multiplication Principle L L L # # # How many possibilities for the 5 th position? 26 * 26 * 10

20. Using the Multiplication Principle L L L # # # How many possibilities for the 6 th position? 26 * 26 * 10 Total number of distinct license plates = 17,576,000

21. Counting the number of ways to arrange songs on a CD: ___ ___ ___ ___ ___ ___ 9108 How many options are there for the 1 st song? 10 songs fit into 6 spots on the demo disk. How many options are there for the 2 nd song? 3 rd song? 4 th song? 5 th song? 6 th song? 576 The “multiplication principle.” When arranging things in order (letters A, B, and C), the total number of possible ways to arrange things is the product of the number of possibilities for each step. of the number of possibilities for each step.

23. Probability “What’s the chance of something happening?” “There is a 100% chance it will rain today.” “There is less than a 5% chance you will be picked.

24. Think about “Numerical Measure” Can probability be equal to 50%? What is the largest number that a probability can be? What is the smallest number that a probability can be? Can there be a (– 20)% chance something will happen?

25. Probability When discussing probability, you can use either “%”, fraction, or the decimal equivalent.  “There is a 40% chance of thunderstorms today.” In mathematics, we convert % to the decimal equivalent or leave it in fraction form.  “The probability of rain today is 0.4.”

26. Your Turn 2. The probability that a student passes the drivers ed. Written test is 62%. The probability that a student passes the driving part of the test is 86%. Draw a tree diagram showing the different outcomes (two branches then two more branches) a. Find the end probability for each of the branches of your diagram. b. Add up all the end probabilities c. What is the probability that the student passes both tests? d. What is the probability that a student passes only one of the tests? *this is problem 2 on your homework

27. Again 3. Every day that you drive to school you break the speed limit. The probability that you won’t get a ticket is 98%. Wahoo! You think that you will never get a ticket! You will make 180 trips to school this year. a.What is the probability that you won’t get any tickets for the whole year? (use percents) b.What is the probability that you will get at least one ticket? (use percents) *this is problem 3 on your homework

28. Vocabulary Review Numerical Measure Experiments Sample Space Outcomes Diagrams Tree Diagrams Multiplication Principal Factorials

29. Homework 10.1 HW 10.1 Basic Probability Story Problems 1-7 – *use tree diagrams to solve problems.