Question

\(25 \%\) of the population are smokers A smoker has 27 times more chances to develop lung cancer than a non smoker A person is diagnosed with lung cancer and the probability that this person is a smoker is \(\frac{k}{10}\), Then the value of \(k\) is____

Answer 1

Correct Answer: 9

We are given the following information: 25% of the population are smokers. A smoker has 27 times the likelihood of developing lung cancer compared to a non-smoker. We need to find the probability \( P(\text{Smoker} \mid \text{Lung Cancer}) = \frac{k}{10} \).

Let's denote:
The probability a person is a smoker: \( P(S) = 0.25 \);
The probability a person is a non-smoker: \( P(S^c) = 0.75 \);
The probability a smoker develops lung cancer: \( P(L \mid S) \);
The probability a non-smoker develops lung cancer: \( P(L \mid S^c) \);
We know \( P(L \mid S) = 27 \times P(L \mid S^c) \).

We aim to find \( k \) such that \( \frac{k}{10} = P(S \mid L) \). By Bayes' theorem:

\( P(S \mid L) = \frac{P(L \mid S) \cdot P(S)}{P(L)} \).

First, calculate \( P(L) \):

\( P(L) = P(L \mid S) \cdot P(S) + P(L \mid S^c) \cdot P(S^c) \).

Let \( x = P(L \mid S^c) \), then \( P(L \mid S) = 27x \).

Substitute in \( P(L) \):
\( P(L) = (27x \cdot 0.25) + (x \cdot 0.75) = 6.75x + 0.75x = 7.5x \).

Now substitute back to find \( P(S \mid L) \):
\( P(S \mid L) = \frac{27x \cdot 0.25}{7.5x} = \frac{6.75x}{7.5x} = \frac{6.75}{7.5} = \frac{9}{10} \).

Thus, \(\frac{k}{10} = \frac{9}{10}\), leading to \( k = 9 \).

Checking the range: the computed \( k = 9 \) is within the specified range of \( 9,9 \).