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Probability I

Probability I. Notes. Probability of A occurring P(A) Sum of all possible outcomes = 1. Sample Space. the collection of all possible outcomes of a chance experiment Roll a die S={1,2,3,4,5,6}. # Of Occurrences of Event #Trials Not rolling a even # E C ={1,3,5}.

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Probability I

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  1. Probability I

  2. Notes Probability of A occurring P(A) Sum of all possible outcomes = 1

  3. Sample Space the collection of all possible outcomes of a chance experiment Roll a die S={1,2,3,4,5,6}

  4. # Of Occurrences of Event #Trials Not rolling a even # EC={1,3,5} Relative Frequency

  5. The long run relative frequency will approach the actual probability as the number of trails increases Coins? 2, 10, 20. The Law of Large Numbers

  6. Event • any collection of outcomes from the sample space • Rolling a prime # E= {2,3,5}

  7. Consists of all outcomes that are notin the event Not rolling a even # EC={1,3,5} P(A) = 1 – P(A) Complement

  8. two events have nooutcomes in common Roll a “2” or a “5” Draw a Black card or a Diamond Mutually Exclusive (disjoint)

  9. two events haveoutcomes in common Draw a Black card or a Spade Not -Mutually Exclusive (Non- disjoint)

  10. Union—Disjoint • the event A orB happening • consists of all outcomes that are in at least one of the two events • Draw a Black card or a Diamond

  11. Union—Disjoint • Draw a Black card or a Diamond • P(B UD) = P(B) + P(D)

  12. Intersection • the event A and B happening • consists of all outcomes that are in both events • Draw a Black card anda 7

  13. Intersection • P(B S) = P(B)•P(S) • Draw a Black card anda 7 U

  14. Union—Not Disjoint • the event A orB happening BUT WE CAN’T Double Count! • Draw a Black card or a 7 • P(B or 7) = P(B) + P(7) – P(B and 7)

  15. Venn Diagrams • Used to display relationships between events • Helpful in calculating probabilities

  16. Venn Diagram Mutually Exclusive / Disjoint events A B

  17. Venn Diagram Not Mutually Exclusive / Non- Disjoint events A B

  18. Venn diagram - Complement of A A

  19. Venn diagram - A and B A B

  20. Stat Cal Com Sci Statistics & Computer Science & not Calculus

  21. Stat Cal Stat Cal Com Sci Com Sci (Statistics or Computer Science) and not Calculus

  22. Two- Way Table • P ( has pierced ears. ) • P( is a male or has pierced ears. ) • P( is a female or has pierced ears )

  23. Rule 1.Legitimate Values For any event E, 0 < P(E) < 1 Rule 2.Sample space If S is the sample space, P(S) = 1 Basic Rules of Probability

  24. Rule 3.Complement • For any event E, • P(E) + P(not E) = 1 • Or • P(not E) = 1 – P(E)

  25. Rule 4.Addition (A or B) • If two events E & F are disjoint, • P(E or F) = P(E) + P(F) • (General) If two events E & F are not disjoint, • P(E or F) = P(E) + P(F) – P(E & F)

  26. Ex 1) A large auto center sells cars made by many different manufacturers. Three of these are Honda, Nissan, and Toyota. Suppose that P(H) = .25, P(N) = .18, P(T) = .14. Are these disjoint events? yes P(H or N or T) = .25 + .18+ .14 = .57 P(not (H or N or T) = 1 - .57 = .43

  27. Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs Flip a Coin and Get Heads. Flip a coin again. P(T) Draw a 7 from a deck. Draw another card. P(8) Independent Independent Not independent

  28. Rule 5.Multiplication If two events A & B are independent, General rule:

  29. The probability that a student will receive a state grant is 1/3, while the probability she will be awarded a federal grant is ½. If whether or not she receives one grant is not influenced by whether or not she receives the other, what is the probability of her receiving both grants?

  30. Suppose a reputed psychic in an extrasensory perception (ESP) experiment has called heads or tails correctly on TEN successive coin flips. What is the probability that her guessing would have yielded this perfect score?

  31. Tree Diagrams Consider flipping a coin twice. What is the probability of getting two heads? Sample Space: HH HT TH TT

  32. Tree Diagrams • Getting Tails Twice

  33. Example: Teens with Online Profiles The Pew Internet and American Life Project finds that 93% of teenagers (ages 12 to 17) use the Internet, and that 55% of online teens have posted a profile on a social-networking site. What percent of teens are online and have posted a profile? 51.15% of teens are online and have posted a profile.

  34. Ex. 3) A certain brand of cookies are stale 5% of the time. You randomly pick a package of two such cookies off the shelf of a store. What is the probability that both cookies are stale? Can you assume they are independent?

  35. Ex 5) Suppose I will pick two cards from a standard deck without replacement. What is the probability that I select two spades? Are the cards independent? NO P(A & B) = P(A) · P(B|A) Read “probability of B given that A occurs” P(Spade & Spade) = 1/4 · 12/51 = 1/17 The probability of getting a spade given that a spade has already been drawn.

  36. Ex. 6) A certain brand of cookies are stale 5% of the time. You randomly pick a package of two such cookies off the shelf of a store. What is the probability that exactly one cookie is stale? P(exactly one) = P(S & SC) or P(SC & S) = (.05)(.95) + (.95)(.05) = .095

  37. Ex. 7) A certain brand of cookies are stale 5% of the time. You randomly pick a package of two such cookies off the shelf of a store. What is the probability that at least one cookie is stale? P(at least one) = P(S & SC) or P(SC & S) or (S & S) = (.05)(.95) + (.95)(.05) + (.05)(.05) = .0975

  38. Rule 6.At least one The probability that at least one outcome happens is 1 minus the probability that no outcomes happen. P(at least 1) = 1 – P(none)

  39. Ex. 7 revisited) A certain brand of cookies are stale 5% of the time. You randomly pick a package of two such cookies off the shelf of a store. What is the probability that at least cookie is stale? P(at least one) = 1 – P(SC & SC) .0975

  40. Dr. Pepper Ex 8) For a sales promotion the manufacturer places winning symbols under the caps of 10% of all Dr. Pepper bottles. You buy a six-pack. What is the probability that you win something? P(at least one winning symbol) = 1 – P(no winning symbols) 1 - .96 = .4686

  41. Warm Up Allergies What is the probability of not having allergies? What is the probability of having allergies if you are a male? Are the events “Female” and “allergies” independent? Justify your answer.

  42. Conditional Probability and Independence Are the events “female” and “right handed” independent?

  43. A probability that takes into account a given condition Rule 7: Conditional Probability

  44. . What is the probability that a randomly selected resident who reads USA Today also reads the New York Times? There is a 12.5% chance that a randomly selected resident who reads USA Today also reads the New York Times.

  45. When performing a random simulation we can use Table D. • Lets say I have a 30% Chance of winning a class lottery. Using Table D…

  46. Probabilities from two way tables Stu Staff Total American 107 105 212 European 33 12 45 Asian 55 47 102 Total 195 164 359 What is the probability that the driver is a student?

  47. Probabilities from two way tables Stu Staff Total American 107 105 212 European 33 12 45 Asian 55 47 102 Total 195 164 359 What is the probability that the driver is staff and drives an Asian car?

  48. Probabilities from two way tables Stu Staff Total American 107 105 212 European 33 12 45 Asian 55 47 102 Total 195 164 359 If the driver is a student, what is the probability that they drive an American car? Condition

  49. Whiteboard Challenge

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