A. Using the information above, find the joint disitrbution of (Drunk Driving, D
ID: 2966700 • Letter: A
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A. Using the information above, find the joint disitrbution of (Drunk Driving, Death). SHow specifically how you applied the formula p(x,y) = p(y|x)*p(x) to obtain all the entries.
Probability of Death when Driving Drunk In the introduction to this chapter you saw a large difference between Pr(death drunk) and Pr(drunk death). How can you convert the statistic "40% of auto fatalities involve a drunk driver" to a probability that someone will die the next time you frive while intorcocated? It's impossible to give a precise answer to this question, but it is possible to arrive at an answer is at least in the same ball park. The first step is to identity an X and a Y. One Variable is the binary indicator of driving while drunk versus driving while sober, and the other is the indicator of whether the trip ends in fatality or no fatality. While it does not technically matter which of these binary variables you call X and which you call Y, Bayes' theorem, as stated previously, starts with knowledge of p(y x), then converts it to p(x y). In this framework, the given information, 40%, is part of p(y x), then converts it to p(x y). So let Y = driving method (drunk or sober), and let X = trip outcome (fatality or non-fatality). The distribution of Y X = fatality, as suggested by the roadside sign, is shown in table 6.17. But if you are planning to drive drunk-which, of course, we do not recommend-you will want the distribution of X, trip outcome, given Y = driving drunk, not vice cersa. (Use what you know to predict what you don't know.) Bayes' theorem as given by Equation 6.14 states p(x y) = p(y x) p(x), where you view x as variable (fatality or non-fatality) and y as fixed (drunk). So you also need the distribution of Y (drunk or sober) given X = non-fatality. It is reasonable to assume that most driving excursions do not end in fatality, so the percentage of non-fatal car trips where the driver is drunk should be approximately the same as the percentage of drivers who are drunk. According to police check-point data, around 1% of drivers are drunk. Thus,a areasonable guess of p(y X = non-fatality) is as given in Table 6.18. Distribution of Drunk Drivers among Trips Ending in a Fatality Distribution of Drunk Drivers among Trips Not Ending in a fatality You now have all the information you need about p(y x) in the expression p(x y) - p(y x) p(x). Once you know p(x), you can plug everything in to get p(x y). How many trips end in fatalities? Auto statistics show that there are around 1.5 fatalities per 100 million vehicle miles traveled. If a typical car excursion is 5 miles, then there are around 1.5 fatalities per 20 million excursions, or an approximate probability of death in an excursion of 1.5/20,000,000 = 0.000000075. This figure translates to 7 or 8 fatalities per 100 million trips. Table 6.19 gies the resulting estimated distribution of X. You want to know Pr(fatality drunl). Taking the relevant information from Tables 6.17 through 6.19, you get table 6.20. Notice that the numbers in the p(Drunk x) column of Table 6.20 do not add to 1.0, nor are they supposed to. Now, in the expression p(x y) prop p(y x) p(x), the term p(y x) p(x) is the product of the last two columns in Table 6.20, as given by Table 6.21. Distribution of Outcomes of Car Trips Distribution of Outcomes of Car Trips along with Conditional Probabilities of Drunken Driving Distribution of Outcomes of Car Trips along with Conditional Probabilities of Drunken Driving and Calculations for Bayes' Theorem infinityExplanation / Answer
given parameters
P(fatal) = 0.000000075 ; P(non fatal)=0.999999925
P(drunk| fatal) = 0.4 ; P(sober | fatal) = 0.6
P(drunk| non fatal)=0.01 ; P(sober| non fatal)= 0.99
To find
P(fatal | drunk) = P(drunk | fatal) * P(fatal) = 0.4 * 0.000000075 = 0.00000003
P(non fatal | drunk ) = P(drunk | non fatal) * P(non fatal) = 0.01 * 0.999999925 = 0.0999999925
P(fatal | sober) = P ( sober | fatal) * P(fatal) = 0.6* 0.000000075 = 0.000000045
P(non fatal | sober) = P(sober| non fatal) * P(non fatal) = 0.99* 0.999999925 = 0.98999992575
B) P( death | driving sober) = 0.000000045 as shown above
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