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Probability

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Probability

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    1. Probability

    2. I am offered two lotto cards: Card 1: has numbers Card 2: has numbers Which card should I take so that I have the greatest chance of winning lotto? Lotto

    3. In the casino I wait at the roulette wheel until I see a run of at least five reds in a row. I then bet heavily on a black. I am now more likely to win. Roulette

    4. Coin Tossing I am about to toss a coin 20 times. What do you expect to happen? Suppose that the first four tosses have been heads and there are no tails so far. What do you expect will have happened by the end of the 20 tosses ?

    5. Coin Tossing Option A Still expect to get 10 heads and 10 tails. Since there are already 4 heads, now expect to get 6 heads from the remaining 16 tosses. In the next few tosses, expect to get more tails than heads. Option B There are 16 tosses to go. For these 16 tosses I expect 8 heads and 8 tails. Now expect to get 12 heads and 8 tails for the 20 throws.

    6. In a TV game show, a car will be given away. 3 keys are put on the table, with only one of them being the right key. The 3 finalists are given a chance to choose one key and the one who chooses the right key will take the car. If you were one of the finalists, would you prefer to be the 1st, 2nd or last to choose a key? TV Game Show

    7. Let’s Make a Deal Game Show You pick one of three doors two have booby prizes behind them one has lots of money behind it The game show host then shows you a booby prize behind one of the other doors Then he asks you “Do you want to change doors?” Should you??! (Does it matter??!) See the following website: http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html

    8. Game Show Dilemma Suppose you choose door A. In which case Monty Hall will show you either door B or C depending upon what is behind each. No Switch Strategy ~ here is what happens

    9. Game Show Dilemma Suppose you choose door A, but ultimately switch. Again Monty Hall will show you either door B or C depending upon what is behind each. Switch Strategy ~ here is what happens

    10. Matching Birthdays In a room with 23 people what is the probability that at least two of them will have the same birthday? Answer: .5073 or 50.73% chance!!!!! How about 30? .7063 or 71% chance! How about 40? .8912 or 89% chance! How about 50? .9704 or 97% chance!

    11. Probability In our discussion of probability Introduce the basic ideas about probabilities: what they are and where they come from simple probability models (genetics) conditional probabilities independent events Baye’s Rule Examine how to calculate probabilities: Using counting methods, tables of counts (not in text) and using properties of probabilities such as independence.

    12. Probability I toss a fair coin (where fair means ‘equally likely outcomes’) What are the possible outcomes? Head and tail ~ This is called a “dichotomous experiment” because it has only two possible outcomes. S = {H,T}. What is the probability it will turn up heads? 1/2 I choose a duck nest at random and observe whether it gets predated or not. What are the possible outcomes? Predated or Not Predated (“Success” and “Failure”) What is the probability of predation? ????? What factors influence this probability? ?????

    13. What are Probabilities? A probability is a number between 0 & 1 that quantifies uncertainty. A probability of 0 identifies impossibility A probability of 1 identifies certainty

    14. Where do probabilities come from?

    15. Probabilities from data or Empirical probabilities What is the probability that a randomly selected starling is female? A random sample n = 67 starlings was taken. 40 of these starlings are female. The estimated probability that a randomly chosen starling will be a female is 40/67 (0.597 or 59.7% chance) Where do probabilities come from?

    16. Subjective Probabilities The probability that there will be another outbreak of ebola in Africa within the next year is 0.1. The probability of snow in the next 24 hours is very high. Perhaps the weather forecaster might say a there is a 80% chance of snow. A doctor may state your chance of successful treatment, e.g. 70% chance of remission. Where do probabilities come from?

    17. Simple Probability Models “The probability that an event A occurs” is written in shorthand as P(A).

    18. Example 1: Hodgkin’s Disease

    19. Example 1: Hodgkin’s Disease

    20. Example 1: Hodgkin’s Disease

    21. Example 1: Hodgkin’s Disease

    22. Example 1: Hodgkin’s Disease

    23. Conditional Probability We wish to find the probability of an event occuring given information about occurrence of another event. For example, what is probability of developing lung cancer given that we know the person smoked a pack of cigarettes a day for the past 30 years. Key words that indicate conditional probability are: “given that”, “of those”, “if …”, “assuming that”

    24. “The probability of event A occurring given that event B has already occurred” is written in shorthand as P(A|B) Conditional Probability

    25. Independence Events A and B are said to be independent if P(A|B) = P(A) and P(B|A) = P(B) i.e. knowing something about the occurrence of B tells you nothing about the occurrence of A.

    26. Example 1: Hodgkin’s Disease

    27. Example 1: Hodgkin’s Disease

    28. Example 1: Hodgkin’s Disease

    29. Example 1: Hodgkin’s Disease

    30. Example 1: Hodgkin’s Disease

    31. Example 1: Hodgkin’s Disease

    32. Example 2: Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)

    33. Example 2: Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)

    34. Example 2: Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)

    35. Example 2: Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)

    36. Example 2: Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)

    37. Example 2: Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)

    38. Building a Contingency Table from a Story Example 3: HIV and Condom Use A European study on the transmission of the HIV virus involved 470 heterosexual couples. Originally only one of the partners in each couple was infected with the virus. There were 293 couples that always used condoms. From this group, 3 of the non-infected partners became infected with the virus. Of the 177 couples who did not always use a condom, 20 of the non-infected partners became infected with the virus.

    39. Let C be the event that the couple always used condoms. (NC be the complement) Let I be the event that the non-infected partner became infected. (NI be the complement) Example 3: HIV and Condom Use

    40. A European study on the transmission of the HIV virus involved 470 heterosexual couples. Originally only one of the partners in each couple was infected with the virus. There were 293 couples that always used condoms. From this group, 3 of the non-infected partners became infected with the virus. Example 3: HIV and Condom Use

    41. Of the 177 couples who did not always use a condom, 20 of the non-infected partners became infected with the virus. Example 3: HIV and Condom Use

    42. What proportion of the couples in this study always used condoms? Example 3: HIV and Condom Use

    43. What proportion of the couples in this study always used condoms? Example 3: HIV and Condom Use

    44. If a non-infected partner became infected, what is the probability that he/she was one of a couple that always used condoms? 4. HIV Example

    45. Example 3: HIV and Condom Use c) In what percentage of couples did the non-HIV partner become infected amongst those that did not use condoms? P(I|NC) = 20/177 = .113 or 11.3% Amongst those that did where condoms? P(I|C) = 3/293 = .0102 or 1.02% What is relative risk of infection associated with not wearing a condom? RR = P(I|NC) / P(I|C) = 11.08 times more likely to become infected.

    46. Example 3: HIV and Condom Use

    47. Relative Risk (RR) and Odds Ratio (OR) Example: Age at First Pregnancy and Cervical Cancer A case-control study was conducted to determine whether there was increased risk of cervical cancer amongst women who had their first child before age 25. A sample of 49 women with cervical cancer was taken of which 42 had their first child before the age of 25. From a sample of 317 “similar” women without cervical cancer it was found that 203 of them had their first child before age 25. Q: Do these data suggest that having a child at or before age 25 increases risk of cervical cancer?

    48. Relative Risk (RR) and Odds Ratio (OR) The ODDS for an event A are defined as Odds for A = _______

    49. Relative Risk (RR) and Odds Ratio (OR) The Odds Ratio (OR) for a disease associated with a risk factor is ratio of the odds for disease for those with risk factor and the odds for disease for those without the risk factor OR = _________________________

    50. Relative Risk (RR) and Odds Ratio (OR)

    51. Relative Risk (RR) and Odds Ratio (OR)

    52. Relative Risk (RR) and Odds Ratio (OR)

    53. Relative Risk (RR) and Odds Ratio (OR)

    54. Relative Risk (RR) and Odds Ratio (OR)

    55. Even though it is inappropriate to do so calculate P(disease|risk status). P(case|Age<25) = 42/245 = .171 or 17.1% P(case|Age>25) = 7/121 = .058 or 5.8% Now calculate the odds for disease given the risk factor status Odds for Disease for 1st Preg. Age < 25 = .171/(1 - .171) = .207 Odds for Disease for 1st Preg. Age > 25 = .058/(1 - .058) = .061 Relative Risk (RR) and Odds Ratio (OR)

    56. f) Finally calculate the odds ratio for disease associated with 1st pregnancy age < 25 years of age. Odds Ratio = .207/.061 = 3.37 This is exactly the same as the odds ratio for having the risk factor (Age < 25) associated with being in the cervical cancer group!!!! Relative Risk (RR) and Odds Ratio (OR)

    57. Relative Risk (RR) and Odds Ratio (OR)

    58. Relative Risk (RR) and Odd’s Ratio (OR) When the disease is fairly rare, i.e. P(disease) < .10 or 10%, then one can show that the odds ratio and relative risk are similar. OR is approximately equal to RR when P(disease) < .10 or 10% chance. In these cases we can use the phrase: “… times more likely” when interpreting the OR.

    59. Relative Risk (RR) and Odds Ratio (OR)

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