Pr(not brown) = 1 – Pr(brown) = 1 – 0.40 = 0.60INDEPENDENCE OF OUTCOMESEXAMPLE Are the eye colors dependent? Suppose I said I randTOTALHIV TestTOTALAngina-freeEXAMPLE An ecologist studied the spatial distribution of treMaplesTOTALPresentAlso, haveTopic (7) – INTRODUCTION TO PROBABILITY 7-1 Topic (7) – Introduction to Probability EXAMPLE You buy a lottery ticket. There are two possible outcomes: you’ll either win money or you won’t. How likely is it you’ll win? Defn: Let E denote an outcome to an experiment. The PROBABILITY that the event E occurs is the likelihood or chance of observing that particular outcome. We denote the probability of an event as P(E) or Pr(E) or Prob(E) EXAMPLE Suppose you’re told the chances of winning the lottery are one in 7.2 million. Hence Pr(winning the lottery with a single ticket) = 17 200 000,, EXAMPLE The probability that a fair die will show a five when tossed is Pr(Five)=16 or 0.1667 or 16.7%.Topic (7) – INTRODUCTION TO PROBABILITY 7-2 Defn: RELATIVE FREQUENCY INTERPRE-TATION: The probability of an event E equals the relative frequency of occurrences of E in an indefinitely long number of identical repetitions of the experiment. Pr(E) = # times E occurs . Total # of repetitions EXAMPLE How do I know that Pr(five) = 0.167 for a 6-sided die? Well if I really want to know this for a die in my hand, I should roll the die 1000s of times and record how many times a five appears. EXAMPLE We have a population whose frequency distribution follows the empirical rule: Y4.253.753.252.752.251.751.25.75.25-.25-.75-1.25-1.75-2.25-2.75-3.25HistogramFrequency2001000Std. Dev = 1.00 Mean = .04N = 1500.00Topic (7) – INTRODUCTION TO PROBABILITY 7-3 So, according to the empirical rule, ~ 68% of the population have Y-values between 0.04 - 1.00 = - 0.96 and 0.04 + 1.00 = 1.04 ~ 95% of the population have Y-values between 0.04 - 2.00 = - 1.96 and 0.04 + 2.00 = 2.04 > 99% of the population have Y-values between 0.04 - 3.00 = - 2.96 and 0.04 + 3.00 = 3.04 What is the probability that a single selection of a unit from this population would result in an observed value greater that 2.04? Using the idea of probability as the relative frequency of occurrence, we have that ~ 2.5% of the population exceeds 2.04. Hence, there is a 2.5% chance that a randomly selected individual will exceed 2.04. Basic Properties of Probability: 1. 0 ≤ Pr(E) ≤ 1 (or 0% ≤ Pr(E) ≤ 100%) 0 1 cannot must occur occurTopic (7) – INTRODUCTION TO PROBABILITY 7-4 2. If outcomes cannot occur simultaneously in a single selection, the probability of observing any of them in a single selection is the sum of their individual probabilities. EXAMPLE Suppose a human population consists of 10% with green eyes, 20% blue eyes, 5% grey eyes, 25% hazel eyes, 40% brown eyes. A person eye color can only be categorized as one of these. What is the probability that a randomly selected person will have either blue or grey eyes? If Pr(blue eyes) = 0.20 and Pr(grey eyes) = 0.05 and one cannot have grey and blue eyes simultaneously, then Pr(blue OR grey eyes) = Pr(blue) + Pr(grey) = 0.25. 3. The probability that an outcome will NOT occur is 1 minus the probability that it will occur. Pr(not E) = Pr(EC) = 1 – Pr(E) e.g. What is the probability that a randomly selected person will have some color other than brown eyes?Topic (7) – INTRODUCTION TO PROBABILITY 7-5 Pr(not brown) = 1 – Pr(brown) = 1 – 0.40 = 0.60 INDEPENDENCE OF OUTCOMES Defn: Two events, E and F say, are said to be INDEPENDENT if the probability of one event occurring is not affected by any knowledge of whether or not the other one has occurred. If the occurrence of one event alters the probability of the other event occurring, the outcomes are said to be DEPENDENT. Notation: The conditional probability is given by Pr(E|F) = Pr ( event E occurs given that event F occurs) INDEPENDENT: Pr(E) = Pr(E|F) DEPENDENT: Pr(E) ≠ Pr(E|F) EXAMPLE Are the eye colors dependent? Suppose I said I randomly selected one person and they had grey eyes. What is the probability they have green eyes given that the event grey eyes occurred?Topic (7) – INTRODUCTION TO PROBABILITY 7-6 Pr(green|grey) = 0 ≠ Pr(green) = 0.10 These events are MUTUALLY EXCLUSIVE EXAMPLE A random sample of 120 college students found that 9 of 61 women had been tested for HIV and 8 out of 59 men had been tested. Female Male TOTAL HIV Test 9 5 14 No Test 52 51 103 TOTAL 61 59 120 Pr(HIV test) = 14/120 = 11.67% Pr(HIV test | female) = 9/61 = 14.8% Pr(HIV test | male) = 5/59 = 8.5% Hence, the probability of a college student getting an HIV test depends on the gender of the student. EXAMPLE A clinical trial involved 307 patients with angina pectoris and two medicines: timolol and a placebo. Patients were randomly assigned (double-Topic (7) – INTRODUCTION TO PROBABILITY 7-7 blind study) to one of the drugs. After 28 weeks they were asked if the angina attacks had ceased. The results are: Timolol Placebo TOTAL Angina-free 44 19 63 Not Angina-free 116 128 244 TOTAL 160 147 307 Pr(Angina-free) = 63/307 = 20.5% Pr(Angina-free | placebo) = 19/147 = 12.9% Pr(Angina-free | timolol) = 44/160 = 27.5% We see that whether a patient is angina-free after 28 weeks on a drug regimen depends on which drug they received. Hence, the two events are dependent. Further examples of independence/dependence: • Two students take an exam. If they work alone, their scores are independent. If they cheat by working together, their scores are not independent. • Lab experiment where animal is rewarded for hitting the right door. Each animal used may be expected to behave independently. However, the same animal gets "smart" overTopic (7) – INTRODUCTION TO PROBABILITY 7-8 time and P(animal picks door with reward) changes over time. Successive trials using the same animal are NOT independent. • Independence
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