[1] Statistical studies have repeatedly shown that about 85% of all bicyclists where helmets. Further studies reveal that about 80% of those wearing helmets do not have head injuries. We do not have any data as to whether those who wear helmets are more safety conscious and therefore have fewer accidents. We do know that about 20% of those who do not wear helmets also experience no head injuries. That's all the data we have. Mr. X was in a bicycle accident and suffered a head injury. (we do not know anything else, so do not try to factor in all sorts of possible scenarios). Given the above information, what is the probability that Mr. X was wearing a helmet? A. First, based on the logic of the problem explain which way you expect the final probability (the posterior) to be revised...up or down? In other words, is it more likely or less likely that he was wearing a helmet? Why? B. Next, demonstrate that final probability using Bayes's formula. What is the p(H|E)? (display the formula and solve) H = the hypothesis that X was wearing a helmet E = head injury The prior, p(H) = .85 Why? What does that figure represent? What is p(~H)? What is p(E|H)? What is p(E|~H)?