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Probability · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQBasic probability concepts and rules10 marksPaper 3~15 min

A medical research team investigates a diagnostic test for a rare disease. The disease affects

0.2\%

of the population. - If a person has the disease, the test returns a positive result

95\%

of the time. - If a person does not have the disease, the test returns a negative result

99.8\%

of the time. Let

D

be the event "a randomly selected person has the disease" and

T

be the event "the test returns a positive result".

(a)

Show that

P(T) \approx 0.00390

. [3 marks]

(b)

Calculate

P(D \mid T)

, the probability that a person who tests positive actually has the disease. [2 marks]

(c)

Evaluate whether this test is suitable for widespread population screening. In your answer, refer to your result from (b), the concept of false positives, and the effect of disease prevalence. The following formulae are provided:

P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad P(A') = 1 - P(A)

P(T) = P(D)\,P(T \mid D) + P(D')\,P(T \mid D')

[5 marks]

Solutions

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