Question

ltis known that 3% ofthe Uk's population carry a certain disease. Atest forthe disease is available Leave 5 which always gives either a positive or a negative result. Given that the individual carries the disease, there is a 98% chance that the test will give a positive result. Given that the individual does not carry the disease, there is a 95% chance that the test will give a negative result. blank a) i) Joey takes the test and gets a positive result Find the probability that he carries the disease. ii) State an assumption that you have made. Hiroshi claims that the test will give the correct result (98% + 95%)-2-96.5% of the time. Is he correct? Show working to support your answer b) c) Joey and Hiroshi disagree about whether the test is effective. By considering your answers to parts a) and b), explain whether you think this is a good test to see if people carry the disease.

Answer 1

a)
X = have the disease
Y = test positive
P(X) = 0.03
P(Y|X) = 0.98
P(Y' | X') = 0.95

P(X|Y) = P(X and Y)/P(Y)
= P(Y|X)P(X) / ( P(Y|X) P(X) + P(Y | X')P(X'))
= 0.98 * 0.03 / (0.98*0.03 + 0.05 * (1-0.03))
= 0.377406

b)
false
this is because number of person who don't have disease is much larger 97%
as compared to 3 % who have disease

c)

we have 37.74% probability that if we test positive then we actually have disease

this is not good test