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**schoolboy46** Problem # 10):

A large research hospital has accumulated statistical data on its patients for an extended period. Researchers have determined that patients who are smokers have an 18% chance of contracting a serious illness such as heart disease, cancer, or emphysema, whereas there is only a .06 probability that a nonsmoker will contract a serious illness. From hospital records, the researchers know that 23% of all hospital patients are smokers, while 77% are nonsmokers. For planning purposes, the hospital physician staff would like to know the probability that a given patient is a smoker if the patient has a serious illness.

I used Bayes Theory for this--and came up with 47% as my answer--just wanting to verify that I am correct on this one.

Personally, have always had trouble with Bayes Theory! Here is how I would do this:

Imagine a hospital with 10000 patients (chosen to give integer numbers). "From hospital records, the researchers know that 23% of all hospital patients are smokers, while 77% are nonsmokers." Okay, so 2300 patients are smokers, 7700 are non-smokers. "Researchers have determined that patients who are smokers have an 18% chance of contracting a serious illness" So .18*2300= 414 so 414 of the smokers have a serious illness. "there is only a .06 probability that a nonsmoker will contract a serious illness." .06*7700= 462 so 462 of the non-smokers have a serious illness.

To find "the probability that a given patient is a smoker if the patient has a serious illness", note that there are a total of 414+ 462= 876 patients with a serious illness, of whom 414 are smokers.

Problem # 12):

The Senate consists of 100 senators, of whom 34 are Republicans and 66 are Democrats. A bill to increase defense appropriations is before the Senate. Thirty- five percent of the Democrats and 70% of the Republicans favor the bill. The bill needs a simple majority to pass. Using a probability tree, determine the probability that the bill will pass.

I think the formula that I want to use here is P(A or B)= P(A) + P (B) - P(AB), but don't know how to take it from here as far as actually setting up a tree...

A= 66 Democrats

B= 34 Republicans

P (A)=0.35

P (B)= 0.70

P(A or B)= 66*0.35 + 34*0.70-(.35*.7)= not adding up??

Problem # 14):

A metropolitan school system consists of three districts— north, south, and central. The north dis-trict contains 25% of all students, the south district contains 40%, and the central district contains 35%. A minimum- competency test was given to all students; 10% of the north district students failed, 15% of the south district students failed, and 5% of the central district students failed.

a. Develop a probability tree showing all marginal, conditional, and joint probabilities.

b. Develop a joint probability table.

c. What is the probability that a student selected at random failed the test?

Don't have a clue where to start based on the one above?

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