Brett and Margo have each thought about murdering their rich Uncle Basil in hope

Question

Brett and Margo have each thought about murdering their rich Uncle Basil in hopes of claiming their inheritance a bit early. Hoping to take advantage of Basil's predilection for immoderate desserts. Brett has put rat poison into the cherries flambé; Margo, unaware of Brett's activities, has laced the chocolate mouse with cyanide. Given the amounts likely to be eaten, the probability of the rat poison being fatal is 0.60; the cyanide, 0.90. Based on other dinners where Basil was presented with the dessert options, we can assume that he has a 50% chance of asking for the cherries flambé, a 40% chance of ordering the chocolate mousse, and a 10% chance of skipping dessert altogether. No sooner are the dishes cleared away than Basil drops dead. In the absence of any other evidence, who should be considered the prime suspect? Brett and Margo have each thought about murdering their rich Uncle Basil in hopes of claiming their inheritance a bit early. Hoping to take advantage of Basil's predilection for immoderate desserts. Brett has put rat poison into the cherries flambé; Margo, unaware of Brett's activities, has laced the chocolate mouse with cyanide. Given the amounts likely to be eaten, the probability of the rat poison being fatal is 0.60; the cyanide, 0.90. Based on other dinners where Basil was presented with the dessert options, we can assume that he has a 50% chance of asking for the cherries flambé, a 40% chance of ordering the chocolate mousse, and a 10% chance of skipping dessert altogether. No sooner are the dishes cleared away than Basil drops dead. In the absence of any other evidence, who should be considered the prime suspect?

Explanation / Answer

We have

Probability of Bret killing uncle Basil =Probability of rat Poisson being fatal*Probability that uncle Basil picks cherris flamable = 0.5*0.6 = 0.3

Probability of Margo Killing uncle Basil =Probability of Uncle Basil picks chocolate mouse*Probability that cyanide is fatal = 0.4*0.9= 0.36

Further probability that Basil dies given that he was not ordered anything =0*0.1 =0 as,chance of skipping desssert altogether =0.1 and probability of dying in this case =0

Probability of Basil being murdered =0.3+0.36 =0.66

Now using the condictional probability

P(Bret kills Uncle Basil given that he is murdered) = Probability of Bret Killing uncle Basil/(Probability of Basil being murdered) =0.3/(0.3+0.36+0) = 0.45

P(Margo kills Uncle Basil given that he is murdered) =Probability of Margo Killing Uncle Basil/(Probability of Basil being murdered) =0.36/(0.3+0.36+0) = 0.54

Since the probability of killing by Margo is greater than the probability of killing by Bret,therefore Margo should be considered as the prime suspect.

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