A certain genetic condition affects 8% of the population in a city of 10,000. Suppose there is a test for the condition that has an error rate of 1% (i.e., 1% false negatives and 1% false positives). Consider the values that would complete the table below. Has condition Does not have condition totals Test positive Test negative Totals What is the probability (as a percentage) that a person has the condition if he or she tests positive? (Round your answer to one decimal place.)

To solve this, we can use Bayes' theorem, which allows us to find the probability of a condition given the test result.

First, let's calculate the expected numbers based on the given percentages.

The total population is 10,000. The percentage of people with the condition is 8%. So, the number of people with the condition is 8% of 10,000, which is 800. The number of people without the condition is 92% of 10,000, which is 9,200. Given the error rate of the test is 1%, we have: 1% false negatives, so 1% of the 800 people with the condition will test negative, which is 8 people. 1% false positives, so 1% of the 9,200 people without the condition will test positive, which is 92 people.

Now, let's fill in the table with these values: ``` Has condition Does not have condition Totals Test positive 792 (800 - 8) 92 (1% of 9,200) 884 Test negative 8 (1% of 800) 9,108 (9,200 - 92) 9,116 Totals 800 9,200 10,000 ```

We are interested in finding the probability that a person has the condition given that they test positive. This is P(Condition | Positive).

Using Bayes' theorem: P(Condition | Positive) = (P(Positive | Condition) * P(Condition)) / P(Positive)

P(Positive | Condition) is the probability of testing positive given the person has the condition. Since there's a 1% false negative rate, the test is 99% accurate for those with the condition, so P(Positive | Condition) = 99%.

P(Condition) is the overall probability of having the condition, which is 8%.

P(Positive) is the probability of testing positive. This is found by taking the total number of positive tests divided by the total population. The total number of positive tests is 884 (from the table), so P(Positive) is 884/10,000, or 8.84%.

Plugging in these values, we get: P(Condition | Positive) = (0.99 * 0.08) / 0.0884 = (0.0792 / 0.0884)

Now, calculate this probability:

P(Condition | Positive) = 0.0792 / 0.0884 ≈ 0.896 (or 89.6%)

However, we want the result as a percentage rounded to one decimal place, therefore:

P(Condition | Positive) ≈ 89.6% (rounded to one decimal place)

So, the probability that a person has the condition if they test positive is approximately 89.6%.

Extra: Bayes' theorem is a method used in statistics to calculate the probability of an event based on prior knowledge of conditions related to the event. In the context of medical testing, it helps to determine the likelihood of a disease given a positive or negative test result. The theorem can be written as:

P(A | B) = (P(B | A) * P(A)) / P(B)

Where: P(A) is the prior probability of event A occurring (in this case, the proportion of people with the condition in the population). P(B) is the total probability of event B occurring (in this case, the proportion of people who test positive for the condition). P(A | B) is the probability of event A occurring given that B is true (the probability that a person has the condition given a positive test result). P(B | A) is the probability of observing event B given that event A is true (the accuracy of the test in correctly identifying the condition).