Question

1. All 400 patients from a heart clinic were categorized according to their NYHA class (NYHA class was described in homework 1) and their age (younger than 40 years, or 40 years or older).

	NYHA I	NYHA II	NYHA III	NYHA IV	Total
Age < 40 years	60	30	20	10	120
Age ≥ 40 years	100	80	60	40	280
Total	160	110	80	50	400

1. Suppose that patients from the heart clinic in problem 2, who were classified as NYHA class II, III or IV, are referred to an imaging clinic for further diagnostic evaluation. The imaging clinic does not see other patients. The imaging clinic determined that 60 of the patients referred to them from the heart clinic are NYHA class IV.

1. What is the probability that a randomly selected patient from the heart clinic will be referred to the imaging clinic?

2. Given the information in problem 2 and problem 3 so far, can you determine how likely a patient would be diagnosed with NYHA IV at the imaging clinic if they had already been diagnosed as NHYA IV at the heart clinic?

3. (*) Now, suppose that the imaging clinic informed us that of the 60 patients that they diagnosed as NYHA IV, 45 of those were also diagnosed as NYHA IV at the heart clinic.
   1. If a patient is diagnosed with NYHA IV at the heart clinic, how likely is it they will be diagnosed as NYHA IV at the imaging clinic?

2. If a patient is diagnosed with NYHA IV at the imaging clinic, how likely is it they would have been diagnosed with NYHA IV at the heart clinic?

Answer 1

Solution

Back-up Theory

If A and B are two events such that probability of B is influenced by occurrence or otherwise of A, then

Conditional Probability of B given A, denoted by P(B/A) = P(B ∩ A)/P(A)……....….(1)

P(A/B) = P(B/A) x {P(A)/P(B)}………………………..………………………..…….(2)

Now, to work out the solution,

Part (a)

Patients classified as Type II, III, and IV at the heart clinic are referred to imaging clinic.

Out of 400 patients at the heart clinic, 240 are classified as Type II, III, and IV.

Combining the above two statements,

probability that a randomly selected patient from the heart clinic will be referred to the imaging clinic

= 240/400

= 0.6 Answer 1

Part (b)

This probability cannot be determined. Answer 2

If we also know how many of the 60 patients the imaging clinic classified as Type IV had been classified as Type IV by the heart clinic, then this probability can be determined.

Part (c)

Now, we know that out of 60 patients the imaging clinic classified as Type IV, 45 had been classified as Type IV by the heart clinic.

Sub-part (i)

If a patient is diagnosed with NYHA IV at the heart clinic, the likely-hood they will be diagnosed as NYHA IV at the imaging clinic

= 45/60

= 0.75 Answer 3

Sub-part (ii)

Just for convenience of presentation and explaining, let A represent the event that a patient is diagnosed with NYHA IV at the imaging clinic and B represent the event a patient is diagnosed with NYHA IV at the heart clinic.

Then, by Answer 3, P(A/B) = 0.75.

What we want to evaluate is: P(B/A), which vide (2)

= P(A/B) x P(B)/P(A)

= 0.75 x (50/400)/(45/240)

= 0.5 Answer 4

DONE