At a certain hospital, 24 patients have lung cancer, 15 patients smoke, and 12 have lung cancer and smoke. Suppose
the hospital contains 120 patients. If a patient chosen at random is known to smoke, what is the probability that the
patient has lung cancer?
If a patient chosen at random is known to smoke, the probability that the patient has lung cancer is
(Type an integer or a simplified fraction.)

Question

UpStudy AI · Accepted Answer

To determine the probability that a patient has lung cancer given that they smoke, we can use conditional probability.

**Given:**
- Total number of patients: 120
- Patients with lung cancer (C): 24
- Patients who smoke (S): 15
- Patients who both smoke and have lung cancer (C ∩ S): 12

**Step-by-Step Calculation:**

1. **Find the Probability of Smoking (P(S)):**
   $$
   P(S) = \frac{\text{Number of patients who smoke}}{\text{Total number of patients}} = \frac{15}{120} = \frac{1}{8}
   $$

2. **Find the Probability of Having Lung Cancer and Smoking (P(C ∩ S)):**
   $$
   P(C \cap S) = \frac{\text{Number of patients who both smoke and have lung cancer}}{\text{Total number of patients}} = \frac{12}{120} = \frac{1}{10}
   $$

3. **Compute the Conditional Probability P(C | S):**
   $$
   P(C | S) = \frac{P(C \cap S)}{P(S)} = \frac{\frac{1}{10}}{\frac{1}{8}} = \frac{8}{10} = \frac{4}{5}
   $$

**Final Answer:**
$$\boxed{\dfrac{4}{5}}$$
The probability that a patient has lung cancer given that they smoke is $$\frac{4}{5}$$.

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