1. Probabilities from Contingency Tables

2. The table below contains data obtained in a study of the relationship between olive oil consumption and cancer of the colon and rectum. Olive Oil Consumption LowMediumHighTotals Cancer Status Colon Cancer3983974301225 Rectal Cancer250241237728 No Cancer1368137714094154 Totals2016201520766107

3. We would first like to calculate the probability that a randomly chosen subject has colon cancer: Out of 6107 subjects, 1225 have colon cancer, so P(Colon Cancer) = 1225/6107 = 0.201 Olive Oil Consumption LowMediumHighTotals Cancer Status Colon Cancer 398397430 1225 Rectal Cancer 250241237728 No Cancer1368137714094154 Totals201620152076 6107

4. Now calculate the probability that a randomly selected subject consumed a medium amount of olive oil: Out of 6107 subjects, 2015 consumed a medium amount, so P(Medium) = 2015/6107 =.330 Olive Oil Consumption LowMediumHighTotals Cancer Status Colon Cancer 3983974301225 Rectal Cancer 250241237728 No Cancer1368137714094154 Totals2016 2015 2076 6107

5. Next, we will calculate the probability that a randomly selected subject has colon cancer and consumed a medium amount of olive oil: Out of 6107 subjects, 397 had colon cancer and consumed a medium amount, so P(Colon Cancer and Medium) = 397/6107 =.065 Olive Oil Consumption LowMediumHighTotals Cancer Status Colon Cancer 398 397 4301225 Rectal Cancer 250241237728 No Cancer1368137714094154 Totals201620152076 6107

6. To calculate the probability that a randomly selected subject has colon cancer or consumed a medium amount of olive oil, we use the General Addition Rule: P(Colon Cancer or Medium) = P(Colon Cancer) + P(Medium) – P(Colon Cancer and Medium) = 1225/6107 + 2015/6107 – 397/6107 = 0.466 Olive Oil Consumption LowMediumHighTotals Cancer Status Colon Cancer 398 397 430 1225 Rectal Cancer 250241237728 No Cancer1368137714094154 Totals2016 2015 2076 6107

7. The events ‘Colon Cancer’ and ‘Rectal Cancer’ are disjoint (mutually exclusive), so, to find the probability that a randomly selected subject has either Colon Cancer or Rectal Cancer, we use the Special Addition Rule: P(Colon Cancer or Rectal Cancer) = P(Colon) + P(Rectal) = 1225/6107 + 728/6107 = 0.320 Olive Oil Consumption LowMediumHighTotals Cancer Status Colon Cancer 3983974301225 Rectal Cancer 250241237728 No Cancer1368137714094154 Totals201620152076 6107

8. Now we will calculate some conditional probabilities. For example, we might want to know the probability that a randomly selected subject has Colon Cancer, given that the subject consumed a Medium amount of olive oil. We are then interested only in the ‘Medium’ column, which corresponds to the given condition. We see that, out of 2015 subjects who consumed a Medium amount of olive oil, 397 have Colon Cancer: P(Colon Cancer | Medium) = 397/2015 = 0.197 Olive Oil Consumption Low Medium HighTotals Cancer Status Colon Cancer 398 397 4301225 Rectal Cancer 250 241 237728 No Cancer1368 1377 14094154 Totals2016 2015 20766107

9. The probability that a randomly selected subject consumed a Medium amount of olive oil given that he has Colon Cancer is different. This time, the given condition is that the subject has Colon Cancer, so we are interested only in the Colon Cancer row. Out of 1225 subjects with Colon Cancer, 397 consumed a Medium amount of olive oil: P(Medium | Colon Cancer) = 397/1225 = 0.324 Olive Oil Consumption LowMediumHighTotals Cancer Status Colon Cancer 3983974301225 Rectal Cancer 250241237728 No Cancer1368137714094154 Totals2016201520766107

10. We might want to know whether the occurrence of Colon Cancer is related to whether or not the subject consumed a Medium amount of olive oil. In other words, are the events ‘Colon Cancer’ and ‘Medium’ independent? To answer this, remember that events A and B are independent if P(A | B) = P(A) and dependent if P(A | B) is different from P(A), so we will compare P(Colon | Medium) with P(Colon). (We could also compare P(Medium | Colon) with P(Medium)). Recall that P(Colon | Medium) =.197, but P(Colon) =.201, so the probability of Colon Cancer is slightly smaller for those who consumed a Medium amount of olive oil than for the entire sample. We can conclude that these two events are Dependent.