Question

1. hospital records show that 14% of all patients are admitted for heart disease, 16% are admitted for cancer (oncology) treatment, and 8% receive both coronary and oncology care. what is the probability that a randomly selected patient is admitted for something other than coronary care? (note that heart disease is a coronary care issue.)

a) 0.84
b) .92
c) .86
d) .78
e) .76

Explanation:

Step1: Define Events

Let \( A \) be the event that a patient is admitted for heart disease (coronary care), and \( B \) be the event that a patient is admitted for cancer (oncology). We know \( P(A) = 0.14 \), \( P(B) = 0.16 \), and \( P(A \cap B) = 0.08 \). We need to find the probability that a patient is admitted for something other than coronary care, i.e., \( P(\overline{A}) \).

Step2: Use Complement Rule

The complement rule states that \( P(\overline{A}) = 1 - P(A) \). Wait, no—wait, actually, we need to check if we misread. Wait, the problem is "probability that a randomly selected patient is admitted for something other than coronary care". Wait, but maybe we need to consider the union? Wait, no, let's re-express. Wait, the problem says "heart disease is a coronary care issue". So we need to find \( P(\text{not } A) \)? Wait, no, maybe the problem is: Wait, the question is "What is the probability that a randomly selected patient is admitted for something other than coronary care?" Wait, but maybe there's a misinterpretation. Wait, let's re-express the events. Wait, maybe the problem is about the union? Wait, no, let's check the numbers. Wait, maybe the problem is: We have \( P(A) = 0.14 \) (coronary), \( P(B) = 0.16 \) (oncology), \( P(A \cap B) = 0.08 \). Wait, but the question is "admitted for something other than coronary care", which is \( P(\overline{A}) \)? But that would be \( 1 - 0.14 = 0.86 \), which is option c. Wait, let's verify.

Wait, the complement of "admitted for coronary care" is "not admitted for coronary care". So \( P(\overline{A}) = 1 - P(A) = 1 - 0.14 = 0.86 \). So that's the probability.

Answer:

c) \( 0.86 \)