Smoking increases the risk of lung problems. A study revealed that \(170\) in \(1000\) males who smoke develop lung complications, while \(120\) out of \(1000\) females who smoke develop lung related problems. In a colony, \(50\) people were found to be smokers of which \(30\) are males. A person is selected at random from these \(50\) people and tested for lung related problems. Based on the given information, answer the following question:
Question: 1
What is the probability that the selected person is a female?
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Always read carefully what group the selection is being made from! Here, the person is picked out of the \(50\) smokers in the colony, so the total sample space size is \(50\). The global statistical data (\(170\) in \(1000\) or \(120\) in \(1000\)) represents conditional probabilities for developing lung complications and is completely irrelevant for this specific introductory sub-question.
If a male person is selected, what is the probability that he will not be suffering from lung problems?
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When a question states "If a male person is selected...", it means the choice has already been restricted to males. You don't need to multiply by the probability of picking a male from the colony. Simply compute the complement directly from the given baseline male group rate: \(1 - \frac{170}{1000} = \frac{830}{1000} = \frac{83}{100}\).
A person selected at random is detected with lung complications. Find the probability that the selected person is a female.
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Bayes' Theorem problems can be cross-verified cleanly by setting up a simple tree diagram or imaginary population numbers! Imagine a scaled group where out of \(500\) total simulated outcomes, \(300\) are male paths and \(200\) are female paths. Then \(300 \times 0.17 = 51\) males get sick, and \(200 \times 0.12 = 24\) females get sick. Total sick people = \(51 + 24 = 75\). Out of those, the female share is exactly \(\frac{24}{75} = \frac{8}{25}\).
A person selected at random is not having lung problems, find the probability that the person is a male.
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When applying Bayes' Theorem for complement events, you can reuse your previous calculation steps efficiently. Since you already know the total sick count in our hypothetical population of \(500\) was \(75\), the total healthy count must be \(500 - 75 = 425\). Out of these, the healthy males account for \(300 - 51 = 249\) people. Thus, the ratio is instantly written down as \(\frac{249}{425}\)!
