Probability and Genetic Predisposition: Should Paul Be Worried?

Let’s analyse a scenario involving a condition, a gene, and a person’s belief about their likelihood of being affected. This problem will help us understand how to use conditional probability to assess risk based on genetic information.

The Scenario:

• P(Condition) = 0.008 (0.8% of the population is affected)
• P(Gene) = 0.901 (90.1% of the population carries the gene)
• P(Neither Gene nor Condition) = 0.0985 (9.85% have neither)

Paul discovers he carries the gene and believes he is very likely to be affected by the condition. Is he correct?

Breaking Down the Problem:

First, let’s define some events:

• C = Event of having the condition
• G = Event of carrying the gene
• C’ = Event of not having the condition
• G’ = Event of not carrying the gene

We want to determine P(C|G), the probability of having the condition given that Paul carries the gene.

Using the Information We Have:

1. Find P(G’):Since P(Gene) = 0.901, then P(Not Gene) = P(G’) = 1 – 0.901 = 0.099
2. Find P(C’ ∩ G’):We are given P(Neither Gene nor Condition) = P(C’ ∩ G’) = 0.0985
3. Find P(G ∩ C’):We know that P(G’) = P(G’ ∩ C) + P(G’ ∩ C’)Therefore, P(G’ ∩ C) = P(G’) – P(G’ ∩ C’)P(G’ ∩ C) = 0.099 – 0.0985 = 0.0005
4. Find P©:We are given P© = 0.008
5. Find P(G ∩ C):We know P© = P(C ∩ G) + P(C ∩ G’)Therefore, P(C ∩ G) = P© – P(C ∩ G’)P(C ∩ G) = 0.008 – 0.0005 = 0.0075
6. Calculate P(C|G):P(C|G) = P(C ∩ G) / P(G)P(C|G) = 0.0075 / 0.901P(C|G) ≈ 0.00832

The Conclusion:

P(C|G) ≈ 0.00832, which is approximately 0.832%. This means that even though Paul carries the gene, his probability of having the condition is still very low, only about 0.832%.

Therefore, Paul’s belief that it is “very likely” that he will be affected by the condition is incorrect. Carrying the gene only slightly increases his risk compared to the general population (0.8%).

This problem demonstrates how important it is to use data and conditional probability to assess risk accurately. Intuition can often be misleading! Make sure you are well prepared for your exams by enrolling in an A Level Maths Revision Course.

This analysis shows that even with the gene, the probability of having the condition remains quite low. Paul should be reassured by these numbers!