House Sales What proportion of the houses that sold for over $600,000 were on the market for less than 30 days? Days on the market Less than 30 days 30 - 90 days More than 90 days Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 Over $600,000 8 12 82 80 19 181 Selling price Total

House Sales What proportion of the houses that sold for over $600,000 were on the market for less than 30 days? Days on the market Less than 30 days 30 - 90 days More than 90 days Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 Over $600,000 8 12 82 80 19 181 Selling price Total

House Sales What proportion of the houses that sold for over $600,000 were on the market for less than 30 days? Days on the market Less than 30 days 30 - 90 days More than 90 days Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 Over $600,000 8 12 82 80 19 181 Selling price Total

House Sales What is the probability a house sold for under $300,000 given that it sold in less than 30 days? Days on the market Less than 30 days 30 - 90 days More than 90 days Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 Over $600,000 8 12 82 80 19 181 Selling price Total

House Sales What is the probability a house sold for under $300,000 given that it sold in less than 30 days? Days on the market Less than 30 days 30 - 90 days More than 90 days Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 Over $600,000 8 12 82 80 19 181 Selling price Total

House Sales What is the probability a house sold for under $300,000 given that it sold in less than 30 days? Days on the market Less than 30 days 30 - 90 days More than 90 days Under $300,000 39 31 15 85 $300,000 - 600,000 35 45 4 84 Over $600,000 8 12 82 80 19 181 Selling price Total

Blood Group Systems (Rh & K) Rh system Kell system Outcome Probability 0.81 x 0.08 0.81 x 0.92 0.19 x 0.08 0.19 x 0.92 K+ K– Rh+ K+ Rh+ K– Rh– K+ Rh– K– 0.08 0.92 Rh+ 0.81 K+ K– 0.08 0.92 0.19 Rh–

Blood Group Systems (K & Rh) Kell system Rh system Outcome Probability 0.08 x 0.81 0.08 x 0.19 0.92 x 0.81 0.92 x 0.19 Rh+ Rh– K+ Rh+ 0. 81 0.19 K+ 0.08 K+ Rh– Rh+ Rh– 0.81 0.19 K– Rh+ 0.92 K– K– Rh–

ELISA: HIV Screening Test 98% true positive and 7% false positive Suppose 1% of the population have HIV Of those that test positive for HIV, what proportion have HIV? 98% of 0.01 Test result 7% of 0.99 Disease status Negative Positive Total Not HIV HIV 0.0098 0.0002 0.01 0.0693 0.9207 0.99 Total 0.0791 0.9209 1

ELISA: HIV Screening Test 98% true positive and 7% false positive Suppose 1% of the population have HIV. Of those that test positive for HIV, what proportion have HIV? Test result Disease status Negative Positive Total Not HIV HIV 0.0098 0.0002 0.01 0.0693 0.9207 0.99 Total 0.0791 0.9209 1

ELISA: HIV Screening Test 98% true positive and 7% false positive Suppose 1% of the population have HIV. Of those that test positive for HIV, what proportion have HIV? 0.0098 / 0.0791 = 0.1239 Test result Disease status Negative Positive Total Not HIV HIV 0.0098 0.0002 0.01 0.0693 0.9207 0.99 Total 0.0791 0.9209 1

ELISA: HIV Screening Test Proportion who don’t have HIV (99%) Proportion who have HIV (1%) Why is the probability of having HIV given that the test is positive so low?

ELISA: HIV Screening Test Positive tests Proportion who don’t have HIV (99%) 98% of 1% Proportion who have HIV (1%) Why is the probability of having HIV given that the test is positive so low?

ELISA: HIV Screening Test True False Positive tests Proportion who don’t have HIV (99%) 98% of 1% Proportion who have HIV (1%) 7% of 99% Why is the probability of having HIV given that the test is positive so low? Many more false positives than true positives

ELISA: HIV Screening Test Disease status Test result HIV and Pos HIV and Neg Not HIV and Pos Not HIV and Neg Outcome Probability 0.98 x 0.01 0.98 0.02 0.07 0.93 Pos | HIV Neg | HIV Pos | Not HIV Neg | Not HIV HIV Not HIV 0.01 0.99 0.02 x 0.01 0.07 x 0.99 0.93 x 0.99 Of those that test positive for HIV, what proportion have HIV? P(HIV Pos) = P(Pos | HIV) x P(HIV)

ELISA: HIV Screening Test Disease status Test result HIV and Pos HIV and Neg Not HIV and Pos Not HIV and Neg Outcome Probability 0.98 x 0.01 0.98 0.02 0.07 0.93 Pos | HIV Neg | HIV Pos | Not HIV Neg | Not HIV HIV Not HIV 0.01 0.99 0.02 x 0.01 0.07 x 0.99 0.93 x 0.99 Of those that test positive for HIV, what proportion have HIV?

ELISA: HIV Screening Test Disease status Test result HIV and Pos HIV and Neg Not HIV and Pos Not HIV and Neg Outcome Probability 0.98 x 0.01 0.98 0.02 0.07 0.93 Pos | HIV Neg | HIV Pos | Not HIV Neg | Not HIV HIV Not HIV 0.01 0.99 0.02 x 0.01 0.07 x 0.99 0.93 x 0.99 Of those that test positive for HIV, what proportion have HIV? 99 . 07 01 98 ) Pos | HIV ( P + ´ =