A more formal way of writing this is: let A and B be 2 events that are mutually exclusive. Then Pr(A or B) = Pr(A) + Pr(B)(Note that Pr(A and B) = 0!) INDEPENDENCE OF EVENTSTOTALHIV TestTOTALAngina-freeEXAMPLE An ecologist studied the spatial distribution of tree species in a wooded area. From a total of 21 acres, he randomly selected 144 quadrats, each 38’ square, and noted the presence or absence of maples and hickories in each quadrat. The results are:MaplesTOTALPresentAlso, haveTopic (6) – Introduction To Probability 6-1 Topic (6) – 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 (6) – Introduction To Probability 6-2 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 Suppose there is a population composed of shell length categories for oysters: 40% are under 40 mm (“spat”), 40% are between 40 and 75mm (“smalls”), 18% are between 75 and 175 mm (“market-size”), and the remainder (2%) are larger than 175 mm. What is the probability that a randomly selected oyster will be market-sized? Suppose 20% of small oysters survive to grow into the market-size group. What is the probability that a small oyster grows to market size?Topic (6) – Introduction To Probability 6-3 EXAMPLE We have a population of values whose frequency distribution follows the empirical rule: So, according to the empirical rule, Y4.253.753.252.752.251.751.25.75.25-.25-.75-1.25-1.75-2.25-2.75-3.25HistogramFr0equency200100Std. Dev = 1.00 Mean = .04N = 1500.00 ~ 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 = 3.04Topic (6) – Introduction To Probability 6-4 What is the probability that a single selection of a unit from this population would result in an observed value greater that 2.04? What is the probability that a single selection falls between 0.04 and 1.04? What is the probability that a single selection falls either less than -1.04 or greater than +2.04?Topic (6) – Introduction To Probability 6-5 Basic Properties of Probability: 1. 0 ≤ Pr(E) ≤ 1 (or 0% ≤ Pr(E) ≤ 100%) 0 1 cannot must occur occur 2. 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 oyster will not be market sized? Actually 2 ways to calculate this?Topic (6) – Introduction To Probability 6-6 3. If outcomes cannot occur simultaneously in a single selection (“mutually exclusive”), the probability of observing any of them in a single selection is the sum of their individual probabilities. EXAMPLE Suppose we have a population consisting of the number of successful fledglings/ nest for a particular species of bird: Number Percent of Nesting Pairs 0 10 1 32 2 31 3 16 4 11 What is the probability of fledging either 1 or 4 birds by a nesting pair during breeding season? A more formal way of writing this is: let A and B be 2 events that are mutually exclusive. Then Pr(A or B) = Pr(A) + Pr(B) (Note that Pr(A and B) = 0!)Topic (6) – Introduction To Probability 6-7 INDEPENDENCE OF EVENTS 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. When two events are dependent, the conditional probability that one has occurred given that the other occurred is: Pr(E|F) = Pr(event E occurs given that event F occurs) = Pr(E and F) / Pr(F) EXAMPLE: Consider the population with a Normal distribution with a mean of 0.04 and a standard deviation of 1.00. What is the probability that a randomly selected value is below -0.96 given that the value is below the mean of 0.04, i.e. find Pr(X < – 0.96 | X < + 0.04).Topic (6) – Introduction To Probability 6-8 It is easy to show that INDEPENDENT: Pr(E) = Pr(E|F) DEPENDENT: Pr(E) ≠ Pr(E|F) EXAMPLE A random sample of 117 college students found that 9 of 61 women had been tested for HIV and 5 out of 56 men had been tested. Female Male TOTAL HIV Test 9 5 14 No Test 52 51 103 TOTAL 61 56 117 Pr(HIV test) = Pr(E) = 14/120 = 11.96% Pr(HIV test | female) = Pr(E|F) = 9/61 = 14.8% Pr(HIV test | male) = Pr(E|M) = 5/56 = 8.9% Hence, the probability of a college student getting an HIV test depends on the gender of the student and so gender and getting an HIV test are not independent events. EXAMPLE A clinical trial involved 307 patients with angina pectoris and two medicines: timolol andTopic (6) – Introduction To Probability 6-9 a placebo. Patients were randomly assigned (double-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%
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