Question

In an experiment to study the dependence of hypertension on smoking habits, the following data were collected on 180 individuals:Smoking StatusNonsmokerModerate SmokerHeavy SmokerHypertension StatusHypertension213630No Hypertension482619What is the probability that a randomly selected individual is experiencing hypertension? Given that a heavy smoker is selected at random from this group, what is the probability that the person is experiencing hypertension? Are the events "hypertension" and "heavy smoker" independent? Give supporting calculations.

Answer 1

First question.

The probability of any event is given by:

`P=\frac{\text{ number of favorable outcomes}}{\text{ number of possible outcomes}}`

In this study we have a total of 87 people with hypertension and 180 people was studied, then we have:

`P=(87)/(180)`

Therefore, the probability of selecting an individual with hypertension is 87/180

Second question.

The conditional probability of event B given that event A already happened is given by:

`P(B|A)=(P(A\cap B))/(P(A))`

For this question let B be the event "The person is experiencing hypertension" and let event A be "The person is a heavy smoker". From the table we notice that a total of 49 individuals are heavy smokers and that 30 individuals are both heavy smokers and experiencing hypertension. Then we have:

`P(B|A)=((30)/(180))/((49)/(180))=(30)/(49)`

Therefore, the probability is 30/49

Third questions.

We know that two events are independent if and only if:

`P(A\cap B)=P(A)P(B)`

Using the events as we did in the previous question we have:

`\begin{gathered} P(A\cap B)=(30)/(180)=(1)/(6)\cong0.17 \\ P(A)P(B)=((87)/(180))((49)/(180))\cong0.13 \end{gathered}`

Clearly the condition is not hold. Which means that the events are dependent