A clinic specializes in testing for a Disease D. The result of the test can be either positive (+ve) or negative (-ve). A study reveals that if a person suffers from the Disease D, the test result in that clinic comes up +ve 80% of the time and negative 20% of the time. If a person is not suffering from the Disease D, the test comes out positive 10% of the time & negative 90% of the time. It is also known among the general population the disease D occurs in 30% of the individuals. If the person tests positive for D in that clinic, the probability that he/she actually suffers from the Disease D is_____(Round off to 2 decimal places).

The correct answer is 0.77.

To find the probability that a person actually suffers from Disease D given that they tested positive, we can apply Bayes' Theorem. Let us first define the events and their given probabilities:
- Let $D$ be the event that a person has the disease, and $D^{\prime }$ be the event that they do not have the disease.
- Let $+$ be the event that the test result is positive.
- Let $-$ be the event that the test result is negative.

From the problem statement, we have the following probabilities:
- Probability of having the disease: $P\left(D\right)=30\%=0.30$
- Probability of not having the disease: $P\left(D^{\prime }\right)=1-P\left(D\right)=0.70$
- Probability of testing positive given that the person has the disease (true positive rate): $P(+|D)=80\%=0.80$
- Probability of testing positive given that the person does not have the disease (false positive rate): $P(+|D^{\prime })=10\%=0.10$

We want to find the conditional probability $P\left(D\right|+)$, which is the probability that the person suffers from the disease given that they tested positive. According to Bayes' Theorem:

$$P\left(D\right|+)=\frac{P(+|D)·P(D)}{P(+|D)·P(D)+P(+\left|D^{\prime }\right)·P\left(D^{\prime }\right)}$$

First, we calculate the total probability of testing positive, which is the denominator of the Bayes' Theorem formula:
$P(+)=(0.80·0.30)+(0.10·0.70)$
$P(+)=0.24+0.07=0.31$

Now, we substitute this back into Bayes' Theorem to find $P\left(D\right|+)$:
$P\left(D\right|+)=\frac{0.24}{0.31}$
$P\left(D\right|+)\approx 0.77419$

Rounding to 2 decimal places, we get:
$P\left(D\right|+)\approx 0.77$