Section 5.1.A Basic Concepts of Probability

1. Section 5.1.A Basic Concepts of Probability Today, you will learn to… > identify the sample space of a probability experiment

2. Sample Space all possible outcomes of a probability experiment Roll a die: { 1, 2, 3, 4, 5, 6 } Flip one coin: { T, H } Gender of one child: { M, F }

3. Event one possible outcome of a trial in an experiment Roll an even number { 2, 4, 6 } Get tails when flipping a coin { H } Draw a queen from a deck of cards {Q, Q, Q, Q}

4. If P(E) = 0 the event is impossible If P(E) = 1 the event is certain 0 < P(E) < 1 or 0% < P(E) < 100%

5. Can probability be… zero? YES - 2%? NO, negative 2.7 ? NO, greater than 1 ¾? YES

6. H T 1 2 3 4 5 6 1 2 3 4 5 6 A probability experiment consists of tossing a coin and then rolling a six-sided die. Identify the sample space. The sample space has 12 outcomes. {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

7. Draw a tree diagram showing the sample space of the gender sequence of a family with 3 children. G B B B B B B G G G G G B G {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}

8. R S S S S S R R R R S R S R SR SR SR SR SR SR SR SR Draw a tree diagram showing the sample space of 4 days of weather (sunny or rainy).

9. R S S S S S R R R R S R S R SR SR SR SR SR SR SR SR List the sample space of 4 days of weather (sunny or rainy). SSSS, SSSR, SSRS, SSRR,

10. R S S S S S R R R R S R S R SR SR SR SR SR SR SR SR {SSSS, SSSR, SSRS, SSRR, SRSS, SRSR, SRRS, SRRR, RSSS,RSSR, RSRS, RSRR, RRSS, RRSR, RRRS, RRRR} 16 different possibilities

11. Simple Event consists of one single outcome Simple: 65 in tall Not Simple: over 6’ tall

12. Decide whether the event is simple or not. the student's age is between 18 and 23 not simple the student’s age is 20 simple

13. Decide whether the event is simple or not. the student scored an 85% simple the student scored a B not simple

14. Let’s Practice!!!!

15. Lesson 5.1.B Types of Probability Theoretical probability Experimental probability Subjective probability

16. Theoretical probability is used when we already have the data we need to find the probability. dice, raffle, cards, coins, etc.

17. 1 52 13 52 You select a card from a standard deck. Find the probability of the following. P(selecting a 7 of diamonds) = = 0.019 2% = P (selecting a diamond) = 0.25 25% = =

18. 12 16 6 16 A bag contains 16 marbles: 10 blue (B), 4 red (R), and 2 green (G). One marble is randomly drawn from the bag. 38% = = P(R or G) = 0.375 = 0.75 = 75% P (not R) =

19. Experimental Probability is based on data collected in an observation or experiment . You actually do an experiment to find the probability

20. 4 200 An insurance company analyst determines that in every 200 claims, 4 are fraudulent. What is the probability that the next claim the company processes is fraudulent? = P(fraudulent)= = 0.02 2%

21. 10 40 A pond contains 3 types of fish.You catch 40 fish and record the type. You catch 13 bluegill, 17 redgill, and 10 catfish. If you throw all of the fish back and catch another fish, what is the probability that it is a catfish? = = P(catfish)= 0.25 25%

22. P(getting heads) = 10 Coin Tossing Probability Experiment 5 = 50% theoretical probability Toss a coin 10 times and count the number of times you get heads. Find the class experimental probability.

23. Law of Large Numbers If you repeat a probability experiment over and over, the experimetnal probability of an event will equal the theoretical probability of the event.

24. 54 1000 Employee data was collected = 0.054 = 5% P(15 - 24 years old) =

25. Subjective probability Results from intuition, educated guesses, and estimates.

26. Subjective probability A doctor may feel that a patient has a 90% chance of a full recovery. A business analyst may predict that there is a 0.25 chance of decreased sales. A weather reporter makes an educated guess that there is a 20% chance of rain today.

27. Theoretical, Experimental, or Subjective? The probability of your phone ringing during class is 0.15 subjective probability because it is most likely based on an educated guess

28. Calculate the probability of the event & its complement. 4 6 E: Pick a red card P(red) = 50% E’: Pick a black P(black) = 50% E: Roll a 3 or greater E’: Roll a 1 or 2 P(>3) = =67% P(<3) = 33%

29. Theoretical, Experimental, or Subjective? The probability that a voter chosen at random will vote republican is 45% statistical probability because it is most likely based on a survey of a sample of voters

30. Theoretical, Experimental, or Subjective? The probability of winning a 1000-ticket raffle with one ticket is 1 in 1000. theoretical probability because you know the number of outcomes and each is equally likely

31. The complement of event E (E’) is the set of all outcomes in a sample space that are NOT included in event E. P(E) + P(E’) = 1 P(E) + P(E’) = 100%

32. 1 6 5 6 3 6 1 2 1 2 P(E’) = P(E) = P(E’) = P(E) = Identify the complement of the event. Give both probabilities. E’: roll a 1,2,3,5,or 6 E: Roll a 4 E: Roll an odd number E’: roll an even

33. Practice Time!!!!

35. Lesson 5.2.A Fundamental Counting Principle Today, we will learn to… > use the Fundamental Counting Principle

36. How many ways I can put together outfits with 3 pairs of pants (jeans, black, and tan) and 4 shirts (white, purple, red, and teal)?

37. white purple red teal white purple red teal white purple red teal 12 ou t f i t s Jeans Black Tan

38. Find the number of ways I can put together a sundae with 3 kinds of ice cream (van, choc, swirl), 3 different toppings (caramel, strawberry, butterscotch) and 3 different crunchy toppings (choc chips, M&Ms, oreo).

39. caramel strawberry butterscotch caramel strawberry butterscotch caramel strawberry butterscotch Choc. Chip M&M Oreo Choc. Chip M&M Oreo Choc. Chip M&M Oreo Choc. Chip M&M Oreo Choc. Chip M&M Oreo Choc. Chip M&M Oreo Choc. Chip M&M Oreo Choc. Chip M&M Oreo Choc. Chip M&M Oreo Van Choc Swirl 27 different sundaes

40. The Fundamental Counting Principle If one event can occur in A ways, another event can occur in B ways, another event can occur in C ways, and so on, then the total number of possible outcomes is A∙B∙C∙… The rule can be extended for any number of events.

41. You are purchasing a new car. You can choose from 4 different manufacturers, 3 different car sizes, and 6 different colors. How many different ways can you select one manufacturer, one car size, and one color? _·_·_ = 4 3 6 72 cars

42. The access code for a car’s security system consist of four digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be used only once and not repeated? 10 __·__·__·__ = 9 8 7 5040

43. The access code for a car’s security system consist of four digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be repeated? 10 __·__·__·__ = 10 10,000 10 10

44. How many license plates can you make if a license plate consists of six letters that cannot be repeated? __·__·__·__·__·__= 25 24 23 22 21 165,765,600 26

45. How many license plates can you make if a license plate consists of six letters that can be repeated? __·__·__·__·__·__= 26 26 26 26 26 308,915,776 26

46. Let’s Practice!!!! 

48. Lesson 5.2.B Permutations Today, we will learn to… > count the number of possible outcomes using a permutation

49. A permutation is an ordered arrangement of objects. The number of different permutations of n objects is n! 5! = 5·4·3·2·1 3! = 3·2·1 2! = 2·1

50. The starting lineup for a baseball team consists of nine players. How many different batting orders are possible using the starting line up? 9·8·7·6·5·4·3·2·1= 9! = 362,880