Question

3 One month of doctors office visits are used to determine the following percentages: 25% of patients had headaches and 40% had the flu. One of the patients is selected at random. Let H represent the event the patient had a headache, and let F represent the even the person had patient had the flu. (a) [I point) Given the information above, what are the smallest and largest values F Explain your reasoning (b) You are also told that if a patient did NOT have a headache, then the probability of the patient having the fu was 40%. Use this additional information to determine the following, showing all of your logic and steps in a clear fashion. i. [1.25 point] P(F n H) ii. [2 points] P(F n H) ii 1.25 point] P(F U H) iv. [2 points] Are the events F and H independent? Explain why or why not.

Answer 1

The given information is:

P(H) = 0.25, P(F) = 0.40

(a)

Minimum:

If F and H are mutually exclusive events then

$P(H\cap F)=0$

Maximum:

If all patient who have headaches have fever also then F will be subset of H in that case

$P(H\cap F)=P(H)=0.25$

(b)

Now we have additional information:

$P(F|\bar{H})=0.40$

(i)

By the complement rule we have,

$P(\bar{H})=1-P(H)=1-0.25=0.75$

So

$P(F\cap \bar{H})=P(F|\bar{H})P(\bar{H})=0.40\cdot 0.75=0.30$

(ii)

By the law of total probability we have:

$P(F)=P(F\cap H)+P(F\cap \bar{H})$

$0.40=P(F\cap H)+0.30$

$P(F\cap H)=0.10$

(iii)

By the addition rule:

$P(F\cup H)=P(F)+P(H)-P(F\cap H)=0.25+0.40-0.10=0.55$

(iv)

Yes they are independent because:

$P(F\cap H)=P(F)P(H)=0.10$