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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.
Answer
To represent the information in the form of a tree diagram, we start with a node for the total population. From this node, we create two branches:
-
Disease (2%)
- Positive Test (95%)
- Negative Test (5%)
-
No Disease (98%)
- Positive Test (3%)
- Negative Test (97%)
This results in a complete tree diagram with six branches, clearly outlining the probabilities.
Step 2
Find the probability that the test is positive.
Answer
To find the probability that the test is positive, we can calculate it using the law of total probability.
egin{align*}
P( ext{Positive Test}) & = P( ext{Disease}) imes P( ext{Positive Test} | ext{Disease}) + P( ext{No Disease}) imes P( ext{Positive Test} | ext{No Disease})
& = 0.02 imes 0.95 + 0.98 imes 0.03
& = 0.0484.
\end{align*}
Step 3
Find the probability that he does not have the disease.
Answer
Given that the test is positive, we can use Bayes' theorem:
egin{align*}
P( ext{No Disease} | ext{Positive Test}) & = \frac{P( ext{Positive Test} | ext{No Disease}) imes P( ext{No Disease})}{P( ext{Positive Test})}
& = \frac{0.03 imes 0.98}{0.0484}
& \approx 0.607.
\end{align*}
Step 4
Comment on the usefulness of this test.
Answer
The test is not very useful because, despite a positive result, there is still a high probability (approximately 60.7%) that the individual does not have the disease. This indicates a high rate of false positives, making the test unreliable for confirming the presence of the disease.