Question

In a particular hospital during Flu season, it was observed that 10% of the patients had Cold. Among the patients with cold, 80% had Flu and among the rest, only 5% had flu. Given that a particular patient had Flu, what is the probability that he had Cold?

Answer 1

Consider x be the total number of patient, i.e. n(S) =x

Assume C is the event of having Cold, F is the event of having flu and N is the event of not having cold.

Since 10% of the patients had Cold,

So, n(C) = 10% of x = 0.1x

Remaining = 90% of x =0.9x

So, n(N)=0.9x

With cold, 80% had Flu,

$n(C\cap F)=80\% \text{ of }0.1x=0.08x$

Among the rest, only 5% had flu.

$n(N\cap F)=5\% \text{ of }0.9x=0.045x$

Thus, total flue patients,

$n(F)=0.08x+0.045x=0.125x$

Now,

$\text{Probability}=\frac{\text{Favorable events}}{\text{Total events}}$

Hence,

$P(F)=\frac{n(F)}{n(S)}=0.125$

$P(C\cap F)=\frac{n(C\cap F)}{n(S)}=0.08$

Therefore, the probability of Cold if it is given that the patient had Flu

$P(\frac{C}{F})=\frac{P(C\cap F)}{P(F)}$

        $=\frac{0.08}{0.125}$

        $=\frac{16}{25}$