The prevalence of diabetes mellitus in a population was found to be 10 per cent. What is the probability that three people selected at random from the population have the disease?
**Core Concept**
The question is testing the concept of probability, specifically the binomial probability distribution. In this context, the probability of three people selected at random from a population having diabetes mellitus is being calculated.
**Why the Correct Answer is Right**
To calculate the probability, we use the binomial probability formula: P(X=k) = (nCk) * (p^k) * ((1-p)^(n-k)), where n is the total number of trials (people selected), k is the number of successful trials (people with diabetes), nCk is the number of combinations of n items taken k at a time, p is the probability of success (prevalence of diabetes), and (1-p) is the probability of failure. In this case, n=3, k=3, p=0.1 (10% prevalence of diabetes), and nCk=1 (since we are selecting 3 people out of 3).
**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because it does not take into account the correct formula for binomial probability. Using the formula P(X=k) = (nCk) * (p^k) * ((1-p)^(n-k)), we can calculate the correct probability.
**Option B:** This option is incorrect because it assumes that the probability of three people selected at random having diabetes mellitus is equal to the prevalence of diabetes in the population. However, the probability of three independent events occurring together is the product of their individual probabilities.
**Option C:** This option is incorrect because it does not account for the correct value of nCk in the binomial probability formula.
**Clinical Pearl / High-Yield Fact**
When calculating the probability of multiple independent events occurring together, we must multiply their individual probabilities. This is a key concept in statistics and is essential for understanding many medical studies and clinical trials.
**Correct Answer: B. 0.001.**