Question

A lab test produces a positive result with 90% probability when the patient is actually sick and with 10% if the patient is healthy. It is known that 15% of the population is sick. (a) What is the joint probability function of patients’ health and test results?

(b) If the test is positive, what is the probability that the patient is actually sick?

(c) The probability you just calculated in part 1b is the _____ probability of ____ given _____.

Answer 1

P[ patient is actually sick ] = 15% = 0.15

P[ patient is not sick ] = 1 - P[ patient is actually sick ] = 1 - 0.15 = 0.85

P[ A lab test produces a positive result | when the patient is actually sick ] = 90% = 0.9

P[ A lab test produces a positive result | when the patient is not sick ] = 10% = 0.1

P[ A lab test produces a positive result and when the patient is actually sick ] = P[ A lab test produces a positive result | when the patient is actually sick ]*P[ patient is actually sick ]

P[ A lab test produces a positive result and when the patient is actually sick ] = 0.9*0.15 = 0.135

P[ A lab test produces a positive result and when the patient is not sick ] = P[ A lab test produces a positive result | when the patient is not sick ]*P[ patient is not sick ]

P[ A lab test produces a positive result and when the patient is not sick ] = 0.85*0.1 = 0.085

P[ A lab test produces a negative result and when the patient is actually sick ] = P[ patient is actually sick ] - P[ A lab test produces a positive result and when the patient is actually sick ]

P[ A lab test produces a negative result and when the patient is actually sick ] = 0.15 - 0.135

P[ A lab test produces a negative result and when the patient is actually sick ] = 0.015

P[ A lab test produces a negative result and when the patient is not sick ] = P[ patient is not sick ] - P[ A lab test produces a positive result and when the patient is not sick ]

P[ A lab test produces a negative result and when the patient is not sick ] = 0.85 - 0.085

P[ A lab test produces a negative result and when the patient is not sick ] = 0.765

a) joint probability function of patients’ health and test results

f(x,y)	Test produces positive result	Test produces negative result	Total
Patient is actually sick	0.135	0.015	0.15
Patient is not sick	0.085	0.765	0.85
Total	0.22	0.78	1

b) If the test is positive, what is the probability that the patient is actually sick?

P[ patient is actually sick | the test is positive ] = P[ A lab test produces a positive result and when the patient is actually sick ] / P[ the test is positive ]

P[ patient is actually sick | the test is positive ] = 0.135/0.22

P[ patient is actually sick | the test is positive ] = 0.6136

(c) The probability you just calculated in part 1b is the conditional probability of patient is actually sick given the test is positive