I need help from these questions. Please attach the problems solving process. As

Question

I need help from these questions. Please attach the problems solving process.

Assume a small town has two fire engines operating independently. The probability that a specific engine is available when needed is 0.94. What is the probability that at least one fire engine is available when needed? Report your answer to four decimal places.

Answer:

In an experiment to study the relationship of hypertension and smoking habits, the following data are collected for 180 individuals:

Smoking Habits

Moderate Smoker

Heavy Smoker

  Hypertension

If one of these individuals is selected at random, find the following probabilities. Report your answer to 4 decimal places.

Consider the following events:

A: The selected individual has hypertension.

B: The selected individual is nonsmoker.

C: The selected individual is moderate smoker.

D: The selected individual is heavy smoker.

Note: Indicate the type of probability as marginal, joint or conditional when asked.

Find the probability that the randomly selected individual has hypertension. Indicate the type of probability.

Find the probability that the randomly selected individual is nonsmoker. Indicate the type of probability.

Find the probability that the randomly selected individual belongs to the category nonsmoker hypertension. Indicate the type of probability.

Given that the selected individual is nonsmoker, what is the probability that the individual has hypertension? Indicate the type of probability.

Based on your answers on part a and d, are smoking habits and hypertension of individuals independent? Provide a mathematical argument?

Consider the events A and B. Are smoking habits and hypertension mutually exclusive events? Provide a mathematical argument to justify your answer.

Find the probability that the randomly selected individual has hypertension or is a nonsmoker.

Consider the events C and D. Are smoking habits mutually exclusive events? Provide a mathematical argument to justify your answer.

Find the probability that the randomly selected individual is moderate or heavy smoker.

What is the probability that a randomly selected individual is neither has a hypertension nor a nonsmoker?

Smoking Habits

Hypertension Nonsmoker

Moderate Smoker

Heavy Smoker

Total

  Hypertension

21 36 30 87
Nonhypertension 48 26 19 93 Total 69 62 49 180

Explanation / Answer

1)probability that the randomly selected individual has hypertension P(A)=87/180=0.4833 ; marginal

2)probability that the randomly selected individual is nonsmoker P(B)=69/180=0.3833 ; marginal

3) probability that the randomly selected individual belongs to the category nonsmoker hypertension

P(A n B)=21/180=0.1167 ; joint probability

4) probability that the individual has hypertension given nonsmokerP(A|B)=21/69=0.3043 ; conditional

5)as P(A)*P(B) is not equal to P(A nB) ; therefore smoking habits and hypertension are not idependent.

6)no as P(A n B) is not equal to 0 ; smoking habits and hypertension are not mutually exclusive events

7)

probability that the randomly selected individual has hypertension or is a nonsmoker

=P(A U B) =P(A)+P(B)-P(A n B) =0.75

8)

Yes ; as P(C n D) =0 ; cause they contain no element in common ; therefore C and D are mutually exclusive

9)

probability that the randomly selected individual is moderate or heavy smoker P(C u D)

=(62+49)/180=0.6167

10)

probability that a randomly selected individual is neither has a hypertension nor a nonsmoker

=P(A u B)' =1-P(A u B) =1-0.75 =0.25

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