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Probability - Part 2 Chapter 3, part 2 Mary Lindstrom (Adapted from notes provided by Professor Bret Larget) February 1, 2004 Statistics 371 Last modified: February 1, 2004 Conditional Probability and Probability Trees It common in biological probability problems for an event to consist of the outcomes from a sequence of possibly dependent chance occurrences. In this case, a probability tree is a very useful device for guiding the appropriate calculations. We have already discussed definitions of probability and events. The following example will illustrate the use of the Conditional Probability Rule, and the Rule of Total Probability. Statistics 371 1 Example The following relative frequencies are known from review of literature on the subject of strokes and high blood pressure in the elderly. • 1. Ten percent of people aged 70 will suffer a stroke within five years; • 2. Of those individuals who had their first stroke within five years after turning 70, forty percent had high blood pressure at age 70; • 3. Of those individuals who did not have a stroke by age 75, twenty percent had high blood pressure at age 70. Statistics 371 2 Example Two questions of interest are: • What is the probability that a 70 year-old patient has high blood pressure? • What is the probability that a 70 year-old patient with high blood pressure will have a stroke within five years? To answer these questions, it is useful to construct a probability tree. Statistics 371 3 Example First let’s define S = {stroke before age 75} Now we can write the statement “Ten percent of people aged 70 will suffer a stroke within five years” as Pr{S} = 0.10 Which tells us right away that Pr{S c} = 1 − Pr{S} = 0.90 Where the notation S c means the complement of the event S. Since S is stroke, S c is no stroke. Statistics 371 4 Rule of complements This is the rule of complements: Pr{E c} = 1 − Pr{E} Where E is any event. Statistics 371 5 Probability tree We can draw a picture of what we know so far Pr{S} = 0.10 Pr{S c} = 0.90 as: 0.1 0.9 Statistics 371 S Sc 6 Example (cont.) “Of those individuals who had their first stroke within five years after turning 70, forty percent had high blood pressure at age 70” becomes Pr{H | S} = 0.40 Where H = {high blood pressure at age 70} The symbol | is read “given” and indicates that 0.40 is a conditional probability. “Of those individuals who did not have a stroke by age 75, twenty percent had high blood pressure at age 70” becomes Pr{H | S c} = 0.20 Statistics 371 7 A Probability Tree for the Example Collecting everything we know: Pr{S} = 0.10 Pr{H | S} = 0.40 Pr{H | S c} = 0.20 We can expand our tree to 0.1 0.9 Statistics 371 0.4 H 0.04 0.6 Hc 0.06 0.2 H 0.18 0.8 Hc 0.72 S Sc 8 Example (cont.) 0.1 0.9 0.4 H 0.04 0.6 Hc 0.06 0.2 H 0.18 0.8 Hc 0.72 S Sc Note that the probabilities show on the branches are conditional. To obtain the unconditional probabilities of the final nodes multiply along the branches. Statistics 371 9 Example (cont.) The questions of interest are: 1. What is the probability that a 70 year-old patient has high blood pressure? We can calculate this as: Pr{H} = Pr{H&S} + Pr{H&S c} = 0.04 + 0.18 = 0.22 This is the Rule of Total Probability. Statistics 371 10 Example (cont.) 2. What is the probability that a 70 year-old patient with high blood pressure will have a stroke within five years? Notice that in this question, the order of conditioning is reversed. We want Pr{S|H} but we have only been given information on Pr{H|S} and Pr{H|S c}. However, there is a formula we can use: Pr{H and S} = 0.04/.22 = 0.182 Pr{S|H} = Pr{H} This is the Conditional Probability Rule. Statistics 371 11