Question

A test correctly identifies a disease in \(95 \%\) of people who have it. It correctly identifies no disease in \(94 \%\) of people who do not have it. In the population, \(3 \%\) of the people have the disease. What is the probability that you have the disease if you tested positive?

Short answer

The probability of having the disease if tested positive is approximately 32.85%.

Step-by-step solution

Identify and Define Probabilities

First, we need to identify the given probabilities and what they represent. - Let \( D \) represent having the disease and \( \overline{D} \) not having the disease.- The probability of having the disease \( P(D) = 0.03 \).- The probability of not having the disease \( P(\overline{D}) = 0.97 \) (since \( 1 - 0.03 = 0.97 \)).- The probability of a positive test given the person has the disease \( P( ext{positive} | D) = 0.95 \).- The probability of a negative test given the person doesn't have the disease \( P( ext{negative} | \overline{D}) = 0.94 \). - Therefore, the probability of a positive test given no disease \( P( ext{positive} | \overline{D}) = 1 - 0.94 = 0.06 \).

Set Up Bayes' Theorem

We need to find the probability that a person has the disease given they tested positive, \( P(D | ext{positive}) \). We use Bayes' Theorem:\[P(D | ext{positive}) = \frac{P( ext{positive} | D) \cdot P(D)}{P( ext{positive})}\]Where \( P( ext{positive}) \) is the total probability of testing positive.

Calculate Total Probability of Testing Positive

The total probability of testing positive, \( P( ext{positive}) \), can be found using:\[P( ext{positive}) = P( ext{positive} | D) \cdot P(D) + P( ext{positive} | \overline{D}) \cdot P(\overline{D})\]Substituting in the known probabilities:\[P( ext{positive}) = 0.95 \cdot 0.03 + 0.06 \cdot 0.97\]\[P( ext{positive}) = 0.0285 + 0.0582\]\[P( ext{positive}) = 0.0867\]

Apply Bayes' Theorem

Now substitute the known values into Bayes' Theorem:\[P(D | ext{positive}) = \frac{0.95 \times 0.03}{0.0867} = \frac{0.0285}{0.0867}\]\[P(D | ext{positive}) \approx 0.3285\]

Interpret the Result

The result \( P(D | ext{positive}) \approx 0.3285 \) tells us the probability of actually having the disease if tested positive is approximately \( 32.85\% \). This lower percentage compared to initial sensitivity is mainly due to the rarity of the disease in the population and the false positives.

Key concepts

Understanding Conditional Probability

Conditional probability is a type of probability that measures the likelihood of an event occurring given that another event has already occurred. This idea is central when dealing with problems like the one in the original exercise, where you want to find out how likely it is to actually have a disease if your test result is positive. In our case, we calculate the probability of having the disease ("you're positive") given that you tested positive. In mathematical terms, if we represent having the disease as \(D\) and testing positive as \(\text{positive}\), conditional probability is expressed as \(P(D | \text{positive})\). This notation is read as "the probability of \(D\) given \(\text{positive}\)." Use Bayes' Theorem to compute this probability, which is crucial for understanding how likely it is that a positive test result indicates an actual case of the disease. Always consider the initial conditions: the sensitivity and specificity of the test, and the prevalence of the condition within the population.

Exploring the False Positive Rate

The false positive rate is a critical concept to grasp when interpreting test results. It represents the probability of the test indicating a positive result, even though the person does not have the disease. In our exercise, this was found by calculating \(P(\text{positive} | \overline{D}) = 0.06\), meaning there's a 6% likelihood that someone who doesn't have the disease might test positive.Understanding the false positive rate is essential because it impacts the reliability of a test result. If the test has a high false positive rate, many healthy individuals could be misdiagnosed as having the disease, leading to unnecessary anxiety and follow-up tests. Thus, when analyzing test results, considering this rate helps in distinguishing between true positives and results that may falsely indicate a disease presence.

Significance of Disease Prevalence

Disease prevalence refers to how widespread a disease is within a specific population at a given time. In the original exercise, it is mentioned that 3% of the population has the disease. This prevalence rate, \(P(D) = 0.03\), directly affects the probability calculations we perform.When a disease is rare, like in this case, there's a higher chance that a positive test result could actually be a false positive due to the relatively small number of people who have the disease. This shows how low prevalence can lead to a higher chance of false positive results, which must be taken into account along with the sensitivity and specificity of the test.In summary, understanding the role of disease prevalence ensures a more accurate interpretation of conditional probabilities and the overall effectiveness of diagnostic tests.