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Chapter 3 Probability

3-1 Fundamentals 3-2 Addition Rule 3-3 Multiplication Rule: Basics 3-4 Multiplication Rule: Complements and Conditional Probability 3-5 Counting Techniques. Chapter 3 Probability. develop sound understanding of probability values used in subsequent chapters

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Chapter 3 Probability

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  1. 3-1 Fundamentals 3-2 Addition Rule 3-3 Multiplication Rule: Basics 3-4 Multiplication Rule: Complements and Conditional Probability 3-5 Counting Techniques Chapter 3Probability

  2. develop sound understanding of probability values used in subsequent chapters develop basic skills necessary to solve simple probability problems Objectives

  3. This chapter tends to be the most difficult one encountered in the course Homework note: Show setup of problem even if using calculator General Comments

  4. Experiment – Action being performed (book uses the word procedure) Event (E) – A particular observation within the experiment Sample space (S) - all possible events within the experiment 3-1 Fundamentals Definitions

  5. P - denotes a probability A, B, ...- denote specific events P (A)- denotes the probability of event A occurring Notation

  6. Let A equal an Event Basic Rules for Computing Probability number of outcomes favorable to event “A” P(A) = total possible experimental outcomes(sample space)

  7. The probability of an impossible event is 0. The probability of an event that is certain to occur is 1. 0  P(A)  1 Probability Limits Certain to occur Impossible to occur

  8. Possible Values for Probabilities Certain 1 Likely 0.5 50-50 Chance Unlikely Impossible 0

  9. Examples: Winning the lottery Being struck by lightning 0.0000035892 1 / 727,235 Typically any probability less than 0.05 is considered unlikely. Unlikely Probabilities

  10. What is the experiment? Roll a die What is the event A? Observe a4 What is the sample space? 1,2,3,4,5,6 Number of outcomes favorable to A is 1. Number of total outcomes is 6. What is P(A)? P(A) = 1 / 6 = 0.167 Example:Roll a die and observe a 4? Find the probability. Similar to #4 on hw

  11. Experiment: toss a coin 3 times Event (A): observe exactly 2 heads Sample Space: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT Note there are 3 outcomes favorable to the event and 8 total outcomes P(A) = 3 / 8 = 0.375 Example:Toss a coin 3 times and observe exactly 2 heads? Test problem Similar to #6 on HW

  12. As a procedure is repeated again and again, the probability of an event tends to approach the actual probability. This is the reason to quit while your ahead when gambling! Law of Large Numbers

  13. The complement of event A, denoted by A, consists of all outcomes in which event A does not occur. P(A) (read “not A”) P(A) Complementary Events or P(Ac)

  14. Property of complementary events P(A) = 1 – P(Ac) Complementary Events Example: If the probability of something occurring is 1/6 what is the probability that it won’t occur?

  15. Example: A study of randomly selected American Airlines flights showed that 344 arrived on time and 56 arrived late, What is the probability of a flight arriving late? Let A = late flight Ac = on time flight P(Ac) = 344 /(344 + 56) = 344/400 = .86 P(A) = 1 – P(Ac) = 1 - .86 = 0.14 Similar to number 11 & 12 on hw

  16. give the exact fraction or decimal or Rounding Off Probabilities • round off the final result to three significantdigits • Examples: • 1/3 is exact and could be left as a fraction or rounded to .333 • 0.00038795 would be rounded to 0.000388

  17. Compound Event – Any event combining 2 or more events Notation – P(A or B) = P (event A occurs or event B occurs or they both occur) General Rule– add the total ways A can occur and the total way B can occur but don’t double count 3-2 Addition Rule Definitions

  18. Formal Addition Rule P(A or B) = P(A) + P(B) - P(A and B) where P(A and B) denotes the probability that A and Bboth occur at the same time. Alternate form P(A B) = P(A) + P(B) – P(A B) Compound Event

  19. Events A and B are mutually exclusive if they cannot occur simultaneously. Definition

  20. Definition Not Mutually Exclusive P(A or B) = P(A) + P(B) – P(A and B) Mutually Exclusive P(A or B) = P(A) + P(B) Total Area = 1 Total Area = 1 P(A) P(B) P(A) P(B) P(A and B) Overlapping Events Non-overlapping Events

  21. Applying the Addition Rule P(A or B) Addition Rule Are A and B mutually exclusive ? Yes P(A or B) = P(A) + P(B) No P(A or B) = P(A)+ P(B) - P(A and B)

  22. Example: P(A) = 2/7 and P(B) = 3/7 , P(A or B) = 5/7, are A and B mutually exclusive? Why? Mutually Exclusive Test question

  23. Let A = choose red marble and B = choose white marble What is the probability of choosing a red marble? P(A) = 3/9 What is the probability of choosing a white marble? P(B) = 4/9 What is the probability of choosing a red or a white? P(B or A) = 3/9 + 4/9 = 7/9 Example:You have an URN with 2 green marbles, 3 red marbles and 4 white marbles Test Questions Why are event A and B mutually exclusive events?

  24. A card is drawn from a deck of cards. What is the probability that the card is an ace or jack? P(ace) + P(jack) = 4/52 + 4/52 = 8/52 What is the probability that the card is an ace or heart? P(ace) + P(heart) – P(ace of hearts) = 4/52 + 13/52 – 1/52 = 16/52 Example:

  25. Experiment: toss a coin 3 times Events (A): observe exactly 0 heads (B): observe exactly 1 head (C):observe exactly 2 heads (D): observe exactly 3 heads Sample Space: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT Find the P(A) + P(B) + P(C) + P(D) Example:Toss a coin 3 times and observe all possibilities of the number of heads

  26. Example:Toss a coin 3 times and observe all possibilities of the number of heads Probability distribution: Table of all possible events along with the probability of each event. The sum of all probabilities must sum to ONE. Note: Events are mutually exclusive

  27. Experiment: roll 2 dice Event (F): observe sum of 5 Sample Space: 36 elements One 2 Two 3’s Three 4’s Four 5’s Five 6’s, One 12 Two 11’s Three 10’s Four 9’s Five 8’s Six 7’s Find the P(F) Example:Roll 2 dice and observe the sum

  28. Example:Roll 2 dice and observe the sum Construct a probability distribution Let’s Try #8 From the HW

  29. Let A = select a man Let B = select a girl P(A or B) = + == 0.781 Contingency Table (Titanic Mortality) Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 45 2223 * Mutually Exclusive * 45 2223 1692 2223 1737 2223

  30. Let A = select a woman Let B = select someone who died. P(A or B) = (422 + 1517 - 104) / 2223 = 1835 / 2223 = 0.825 Contingency Table (Titanic Mortality) Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 45 2223 * NOT Mutually Exclusive * Very similar to test problem

  31. How could you define a probability distribution for this data? Contingency Table (Titanic Mortality) Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 45 2223

  32. Complementary EventsP(A) & P(Ac) • P(A) and P(Ac) are mutually exclusive • P(A) + P(Ac) = 1 (this has to be true) • P(A) = 1 - P(Ac) • P(Ac) = 1 – P(A)

  33. Venn Diagram for the Complement of Event A Total Area = 1 P (A) P (A) = 1 - P (A)

  34. Notation:P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial) Formal Rule P(A and B) = P(A) • P(B) if independent (with replacement) P(A and B) = P(A) • P(B A) if dependent (without replacement) 3-3 Multiplication Rule Definitions will define later

  35. Independent Events Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. Dependent Events If A and B are not independent, they are said to be dependent. Definitions

  36. 1 1 1 5 2 10 Tree Diagram of Test Answers Ta Tb Tc Td Te Fa Fb Fc Fd Fe a b c d e a b c d e T F P(T) = P(c) = P(T and c) =

  37. P (both correct) = P (T and c) 1 10 1 1 = • 2 5 Multiplication Rule INDEPENDENT EVENTS

  38. Choose 2 marbles from an URN with 3 red marbles and 3 white marbles Dependent – choose the 1st marble then choose the 2nd marble Independent– choose the 1st marble, replace it, then choose the 2nd marble Independence vs. Dependence

  39. P(B A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B A as “B given A”). = given Notation for Conditional Probability

  40. Let A = choose red marble and B = choose white marble What is the probability of choosing a red marble? P(A) What is the probability of choosing a white marble? P(B) If two are chosen find the probability of choosing a white on the a second trial given a red marble was chosen 1st. P(B A) Assume the 1st marble is replaced {independent} Assume the 1st marble is not replaced {dependent} Example:You have an URN with 3 red marbles and 4 white marbles Test Questions

  41. Let’s change things a bit…. If two are chosen and we want to find the probability of choosing a white then choosing a red marble. So if we let:A = choose red 1st and B = choose white 2nd then we need to find P(A and B) The problem here is that calculating this probability depends on what happens on the first draw. We need a rule that helps us with this. Example:You have an URN with 3 red marbles and 4 white marbles

  42. P(A and B) = P(A) • P(B A) If A and B are independent events, P(B A) is really the same as P(B). Will see this in the next section. Formal Multiplication Rule

  43. Applying the Multiplication Rule P(A and B) Multiplication Rule Are A and B independent ? Yes P(A and B) = P(A) • P(B) No P(A and B) = P(A) • P(B A)

  44. Let A = choose red marble and B = choose white marble If two are chosen find the probability of choosing a red then choosing a white marble. In other words findP(A and B) Assume the 1st marble is replaced {independent} P(A and B) = P(A) • P(B) Assume the 1st marble is not replaced {dependent} P(A and B) = P(A) • P(B A) Test Questions Example:You have an URN with 3 red marbles and 4 white marbles Use as an example for #6

  45. Let A = choose red B = choose white C = choose green Find the following: P(Ac), that is find P(not red) P(A or B) If two marbles chosen what is the probability that you choose a white marble 2nd when a red marble was chosen first. This is, find P(B A) If two marbles are chosen, find P(A and C) with replacement If two marbles are chosen, find P(A and C) without replacement Class Assignment – Part IYou have an URN with 3 red marbles, 7 white marbles, and 1 green marble Very Similar to test question #1

  46. Class Assignment – Part II

  47. Mutually Exclusive Events P(A or B) = P(A) + P(B) Independent Events P(A and B) = P(A) • P(B) Example:if P(A) = .3, P(B)=.4, P(A or B)=.7, and P(A and B) = .12, what can you say about A and B? Mutually Exclusive vs. Independent Events Note: Test Question

  48. Select two Find P(2 women) = 422/2223 x 421/2222 Find P(2 that died) = 1517/2223 x 1516/2222 Select one Find P(woman and died) = 104/2223 Find P(Boy and survived) = 29 / 2223 Contingency Table (Titanic Mortality) Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 45 2223

  49. Probability of “at least one” More on conditional probability Test for independence 3-4 Topics

  50. ‘At least one’ is equivalent to ‘one or more’. The complement of getting at least one item of a particular type is that you get no items of that type. Probability of ‘At Least One’ If P(A) = P(getting at least one), then P(A) = 1 - P(Ac) where P(Ac) = P(getting none)

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