Introduction to Genetic Inheritance Probability

Understanding the likelihood of inheriting a genetic disease is crucial in both medical and public health contexts. Specifically, if 6.5% of the population has a particular disease and a child has a 13% chance of inheriting it if the father has the disease, what is the probability that a child will inherit the disease when the father does not have it? This article delves into the mathematical principles behind such calculations.

Understanding the Key Factors

To begin, let's define our key terms and the given data:

PD Probability that a person has the disease 0.065 (or 6.5%) PDF Probability that the child has the disease given that the father has the disease 0.13 (or 13%) PDFc Probability that the child has the disease given that the father does not have the disease (what we are trying to find)

Using the Law of Total Probability

The law of total probability is a fundamental principle used to determine the probability of an event based on different scenarios. In this context, it helps us calculate the overall probability of a child inheriting the disease.

We denote the event that the father has the disease as (F) and the event that the father does not have the disease as (F^c). The overall probability (P(D)) can be expressed as:

(P(D) PDF cdot P(F) PDF^c cdot P(F^c))

Where:

(P(F) 0.065) (6.5%) (P(F^c) 1 - P(F) 1 - 0.065 0.935)

Step-by-Step Calculation

Substituting the known values into the equation, we have:

(0.065 0.13 cdot 0.065 PDF^c cdot 0.935)

The first term is:

(0.13 cdot 0.065 0.00845)

Substituting this value back into the equation:

(0.065 0.00845 PDF^c cdot 0.935)

Rearranging to solve for (PDF^c):

(PDF^c cdot 0.935 0.065 - 0.00845)

Calculating the right side:

(0.065 - 0.00845 0.05655)

Now, we can calculate (PDF^c):

(PDF^c frac{0.05655}{0.935} approx 0.0605)

Therefore, the probability that a child will inherit the disease when the father does not have it is approximately 6.05%.

Assumptions and Further Analysis

Two key assumptions guide our analysis:

There are only two generations: the first generation includes fathers who have the disease (6.5% of them), and the second generation may inherit the disease. The disease does not render the patient dead or infertile, meaning 6.5% of the population in question has a father who had the disease.

From these assumptions, we can further deduce that 13% of the second generation will contract the disease if their fathers had it. This means 0.845 (or 84.5%) of the second generation has the disease and had a father who had it. If the overall percentage must be 6.5%, then 6.5 - 0.845 5.655 of the second generation must be contributed by the remaining 93.5% of the population. Therefore, 6.055% of the population that does not have a father who had the disease will inherit the disease.

Conclusion

The detailed calculation and assumptions provide a clear insight into understanding the probability of inheriting a genetic disease. By applying the law of total probability and making reasonable assumptions, we can accurately estimate the likelihood of disease inheritance from one generation to the next.