1. Probability Part 1 – Fundamental and Factorial Counting Rules

2. Probability Warm-up – 1) 2/3 x 6/7 = 2) 5 4 /6 4 = 3) 4/13 + ¼ + ½ - 7/12 =

3. Probability Agenda Warm-up Objective – To introduce the concept of probability and the counting rules Summary Homework

4. Idea of Probability Probability is the science of chance behavior Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run this is why we can use probability to gain useful results from random samples and randomized comparative experiments

5. Probability There are two ways of looking at probability: Random Chance Probability- Events seem to happen in a random way Subjective Probability – Skill of the “player” effects the outcome

6. Probability History During the mid-1600s,a professional gambler named Chevalier de Méré made a considerable amount of money on a gambling game. He would bet unsuspecting patrons that in four rolls of a die, he could get at least one 6. He was so successful at the game that some people refused to play. He decided that a new game was necessary to continue his winnings. By reasoning, he figured he could roll at least one double 6 in 24 rolls of two dice, but his reasoning was incorrect and he lost systematically. Unable to figure out why, he contacted a mathematician named Blaise Pascal (1623–1662) to find out why. Pascal became interested and began studying probability theory. He corresponded with a French government official, Pierre de Fermat (1601–1665), whose hobby was mathematics. Together the two formulated the beginnings of probability theory.

7. Outcomes Before calculating probability- Need to understand the number of ways an event can occur This involves counting rules.

8. Probability probability experiment A probability experiment is a process that leads to well-defined results called outcomes. outcome An outcome is the result of a single trial of a probability experiment. A sample space is the set of all possible outcomes of a probability experiment.

9. Tree Diagrams tree diagram A tree diagram is a device used to list all possibilities of a sequence of events in a systematic way.

10. - Tree Diagrams - Example Suppose a sales person can travel from Boston to New York and from New York to Philadelphia by plane, train, or automobile. Display the information using a tree diagram.

11. - Tree Diagrams - Example Philadelphia Boston New York Plane Train Auto Train Plane Train Auto Plane Auto Train Plane Plane, Auto Plane, Train Plane, Plane Train, Auto Train, Train Train, Plane Auto,Auto Auto, Train Auto, Plane

12. Tree Diagram Thus – by counting the number of stems at the end of the tree – you determine the number of different ways the sequence of events can occur.

13. Fundamental Counting Rule We can also calculate the number of ways by using: Fundamental Counting Rule : n k 1 k 2 k 3 k 1  k 2  k 3  k n Fundamental Counting Rule : In a sequence of n events in which the first one has k 1 possibilities and the second event has k 2 and the third has k 3, and so forth, the total possibilities of the sequence will be k 1  k 2  k 3  k n or  k i i= 1 n

14. Tree Diagrams Example – You have 3 pants, 4 t-shirts, 3 hats, and 2 sneakers. Using a tree diagram, how many different outfits can be created?

15. Example You have 3 pants, 4 t-shirts, 3 hats, and 2 sneakers. Using the fundamental counting rule, how many different outfits can be created? 3 x 4 x 3 x 2 = 72

16. Fundamental Counting Rule Employees of SHS are to be issued special coded identification cards. The card consists of 4 letters of the alphabet. Each letter can be used up to 4 times in the code. How many different ID cards can be issued?

17. Fundamental Counting Rule Since 4 letters are to be used, there are 4 spaces to fill ( _ _ _ _ ). Since there are 26 different letters to select from and each letter can be used up to 4 times, then the total number of identification cards that can be made is 26  26  26  26  = 456,976.

18. Fundamental Counting Rule The digits 0, 1, 2, 3, and 4 are to be used in a 4-digit ID card. How many different cards are possible if repetitions are permitted? Solution: Solution: Since there are four spaces to fill and five choices for each space, the solution is 5  5  5  5 = 5 4 = 625.

19. Fundamental Counting Rule Variation for n events each occurring the same number of times (k) No. of ways = k n

20. Fundamental Counting Rule 1) Create a tree diagram to illustrate going from home to school to work and back home by either walking, by bike, or by car. How many different ways can you make the trip? 2) If you had 3 math books, 4 English books, 2 social studies, and 3 science books. How many different ways could they be arranged on your shelf. 3) A license plate in MA contains 6 characters – 4 numbers followed by 2 letters. Assuming repeats are possible, how many different license plates are possible?

21. Answers Problem 1 Home Car Bike Walk School Car Bike Walk Car Bike Walk Car Bike Walk Work Home Car Bike Walk Car Bike Walk Car Bike Walk Car Bike Walk Car Bike Walk Car Bike Walk Car Bike Walk Car Bike Walk Car Bike Walk

22. Answers Problem 2 3 x 4 x 2 x 6 = 144 possible arrangements Problem 3 10 x10 x 10 x 10 x 26 x 26= 6,760,000 license plates

23. Counting Rules Summary Tree Diagrams Fundamental Counting Rule Factorial Counting Rule

24. Counting Rules Homework Handout #1