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 . Then the value of is _______. [2023]

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Solution

Q.
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 _______. [2023]

Ans.
(9)

Let number of smokers be $E_{1}$ . So, $P (E_{1}) = \frac{1}{4}$ .

The number of non-smokers be $E_{2}$ . So, $P (E_{2}) = \frac{3}{4}$ .

Let E denote persons diagnosed with lung cancer.

Let $P (E / E_{2}) = x, P (E / E_{1}) = 27 x$

Now, $P (E_{1} / E) = \frac{P (E_{1}) P (E / E_{1})}{P (E)}$

$= \frac{\frac{1}{4} \times 27 \times x}{\frac{1}{4} \times 27 \times x + \frac{3}{4} \times x} = \frac{27}{30} = \frac{9}{10}$

So, $k = 9$

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