A disease is known to be present in 2% of a population. A test is developed to help determine whether or not someone has the disease. Given that a person has the disease, the test is positive with probability 0.95. Given that a person does not have the disease, the test is positive with probability 0.03. (a) Draw a tree diagram to represent this information. A person is selected at random from the population and tested for this disease. (b) Find the probability that the test is positive. A doctor randomly selects a person from the population and tests him for the disease. Given that the test is positive, (c) find the probability that he does not have the disease. (d) Comment on the usefulness of this test.

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A disease is known to be present in 2% of a population - Edexcel - A-Level Maths Statistics - Question 1 - 2008 - Paper 2

Question 1

A disease is known to be present in 2% of a population. A test is developed to help determine whether or not someone has the disease. Given that a person has the di... show full transcript

Worked Solution & Example Answer:A disease is known to be present in 2% of a population - Edexcel - A-Level Maths Statistics - Question 1 - 2008 - Paper 2

Step 1

Draw a tree diagram to represent this information.

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Answer

To represent the situation as a tree diagram:

Start with a point representing the total population.
Branch out to two paths: one for having the disease (2% or 0.02) and one for not having the disease (98% or 0.98).
From the 'Disease' branch, create two further branches: one for a positive test (95% or 0.95) and one for a negative test (5% or 0.05).
From the 'No Disease' branch, create two branches: one for a positive test (3% or 0.03) and one for a negative test (97% or 0.97).
The final tree diagram visualizes the probabilities as follows:

                      +--- Disease (0.02)
           +---
           |          +--- Positive Test (0.95)
           |--- +---
                      +--- Negative Test (0.05)
           |          
           |          +--- No Disease (0.98)
           +---
                      +--- Positive Test (0.03)
           |          
                      +--- Negative Test (0.97)

This diagram accurately reflects the disease prevalence and the test outcomes.

Step 2

Find the probability that the test is positive.

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Answer

To find the probability that the test is positive, we can use the law of total probability:

$P(Positive Test) = P(Disease) imes P(Positive | Disease) + P(No Disease) imes P(Positive | No Disease)$

Substituting in the given values:

$P(Positive Test) = (0.02 imes 0.95) + (0.98 imes 0.03)$

Calculating this gives:

$P(Positive Test) = 0.019 + 0.0294 = 0.0484$

Thus, the probability that the test is positive is approximately 0.0484.

Step 3

Find the probability that he does not have the disease.

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Answer

Given that the test is positive, we use Bayes' theorem to find the probability that the person does not have the disease:

$P(No Disease | Positive Test) = \frac{P(Positive Test | No Disease) imes P(No Disease)}{P(Positive Test)}$

We already calculated $P(Positive Test)$ and have:

$P(Positive | No Disease) = 0.03$
$P(No Disease) = 0.98$

Now substituting in:

$P(No Disease | Positive Test) = \frac{0.03 \times 0.98}{0.0484}$

This simplifies to:

$P(No Disease | Positive Test) = \frac{0.0294}{0.0484} \approx 0.607$

Therefore, the probability that the person does not have the disease given that the test is positive is approximately 0.607.

Step 4

Comment on the usefulness of this test.

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Answer

The usefulness of the test can be evaluated based on its probability of correctly identifying individuals who do not have the disease despite a positive test result.

Given that approximately 60.7% of individuals who test positive do not have the disease, this indicates that the test has a relatively high false positive rate. This suggests that the test is not very useful for determining the presence of the disease in the population, as a significant proportion of positive results do not correlate with actual disease presence.

In summary, while the test has high sensitivity (95%) for those with the disease, its low specificity (3% positive test for those without the disease) makes it less reliable, leading to potential misdiagnoses.

A disease is known to be present in 2% of a population - Edexcel - A-Level Maths Statistics - Question 1 - 2008 - Paper 2

Worked Solution & Example Answer:A disease is known to be present in 2% of a population - Edexcel - A-Level Maths Statistics - Question 1 - 2008 - Paper 2

Draw a tree diagram to represent this information.

Find the probability that the test is positive.

Find the probability that he does not have the disease.

Comment on the usefulness of this test.

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