91Ó°ÊÓ

Here given that out of 100,000 people, one person has a type of cancer.

If a test is applied to a person who has the type of cancer, then there is a 95% chance to get a positive. And a 5% chance to get the test result negative.

And if the test is applied to a fine person, then there is a 95% chance to get a negative. And a 5% chance to get the test result positive.

Let us consider the event C, that which a person has that type of cancer. And P be the event that the test has a positive reaction.

Now the probability that a person is affected by the cancer\(P\left( C \right) = \frac{1}{{100000}}\)

The probability that the result is positive given that the person won’t have the cancer is,\(P\left( {P|{C^c}} \right) = 0.05\)and the probability of the result is positive and the person has the cancer is,\(P\left( {P|C} \right) = 0.95\)

And similarly,\(P\left( {{P^c}|C} \right) = 0.5\)

Now we have to compute the probability that a randomly selected person has the type of cancer given that the test result is positive for the person.

That is,

\(\begin{aligned}{}P\left( {C|P} \right) &= \frac{{P\left( C \right)P\left( {P|C} \right)}}{{P\left( C \right)P\left( {P|C} \right) + P\left( {{C^c}} \right)P\left( {P|{C^c}} \right)}}\\ &= \frac{{\frac{1}{{100000}} \times 0.95}}{{\left( {\frac{1}{{100000}} \times 0.95} \right) + \left\{ {\left( {1 - \frac{1}{{100000}}} \right) \times 0.05} \right\}}}\\ &= 0.00018\end{aligned}\)

So, there is a very small probability, about 0.018% of chance that the person has the cancer