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The ancient gene

Suppose there is a gene that some people have that allows them to operate ancient, not yet discovered machines, the so called 'ancient gene'. Obviously it is important to know who has the gene so that they can be properly trained on those mythical machines!
Genesplicer Inc has developed a simple test for the ancient gene. It turns out that about 0.01% of the population has the gene. Unfortunately the simple test is not perfect and in one in 10 000 cases the test comes back with the wrong answer.

Heidi goes to the doctor to get tested. The test comes back positive which means that she probably has the gene.

What is the probability that Heidi actually has the gene?

Hint: its much lower than you think! Assume that 10 000 people take the test, what would the outcome be on average? How many positive test results would you expect?

This problem was listed in a paper that described how many doctors are unable to correctly interpret results, especially when statistics are involved. I cannot find it again, so if you know the reference please let me know. The original problem had no reference to a well known Sci-Fi TV series :-)

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