1. Bayes Theorem

2.  The Gambler’s Fallacy ◦ Is assuming that the odds of a single truly random event are affected in any way by previous iterations of the same or other truly random event.

3.  The Gambler’s Fallacy ◦ Is assuming that the odds of a single truly random event are affected in any way by previous iterations of the same or other truly random event.  Ignoring the Law of Large Numbers ◦ Is assuming there must be other explanations for very improbable events.

4.  a priori probability: ◦ The sort of probability achieved by dividing the number of desired outcomes vs. the total number of outcomes. ◦ Applies to random events.  Statistical probability: ◦ The frequency at which a given event is observed to occur. ◦ Applies to events that are not truly random.

5.  What is the a priori probability (expressed as a percent) of a batter in baseball getting a hit in one at-bat?

6. ◦ 50%

7.  What is the a priori probability (expressed as a percent) of a batter in baseball getting a hit in one at-bat? ◦ 50%  What is the statistical probability (expressed as a percent) of a major league batter getting a hit in one at-bat?

8.  What is the a priori probability (expressed as a percent) of a batter in baseball getting a hit in one at-bat? ◦ 50%  What is the statistical probability (expressed as a percent) of a major league batter getting a hit in one at-bat? ◦ 25.4%

9. Wendy has tested positive for colon cancer. Colon cancer occurs in.3% of the population (.003 statistical probability) If a person has colon cancer, there is a 90% chance that they will test positive (.9 statistical probability of a true positive) If a person does not have colon cancer, then there is a 3% chance that they will test positive (3% statistical probability of a false positive) Given that Wendy has tested positive, what is the statistical probability that she has colon cancer?

10.  The correct answer is 8.3%

11.  Most people (including many doctors) assume that the chances are much better than they really are that Wendy has colon cancer. The reason for this is that people tend to forget that a test must be absurdly specific to give a high probability of having a rare condition.

12. h = the hypothesis e = the evidence for h Pr(h) = the statistical probability of h Pr(e|h) = the true positive rate of e as evidence for h Pr(e|~h) = the false positive rate of e as evidence for h Pr(h|e) = Pr(h) * Pr(e|h) [Pr(h) * Pr(e|h)] + [Pr(~h) * Pr(e|~h)]

13. h~hTotal eTrue Positives False Positives Pr(e)*Pop. ~eFalse Negatives True Negatives Pr(~e)*Pop. TotalPr(h)*Pop.Pr(~h)*Pop. Pop. = 10^n n = sum of decimal places in two most specific probabilities.

14. h~hTotal e= Pr(e|h) * [Pr(h)*Pop. ] = Pr(e|~h) * [Pr(~h)*Po p.] Total of this row ~e= below - above Total of this row TotalPr(h)*Pop.Pr(~h)*Pop. Pop.

15. h~hTotal e= Pr(e|h) * [Pr(h)*Pop. ] = Pr(e|~h) * [Pr(~h)*Po p.] Total of this row ~e= below - above Total of this row TotalPr(h)*Pop.Pr(~h)*Pop. Pop.

16. has CC~ have CCTotal e= Pr(e|h) * [Pr(h)*Pop. ] = Pr(e|~h) * [Pr(~h)*Po p.] Total of this row ~e= below - above Total of this row TotalPr(h)*Pop.Pr(~h)*Pop. Pop.

17. has CC~ have CCTotal tests positive = Pr(e|h) * [Pr(h)*Pop. ] = Pr(e|~h) * [Pr(~h)*Po p.] Total of this row ~ test positive = below - above Total of this row TotalPr(h)*Pop.Pr(~h)*Pop. Pop.

18. has CC~ have CCTotal tests positive = Pr(e|h) * [Pr(h)*Pop. ] = Pr(e|~h) * [Pr(~h)*Po p.] Total of this row ~ test positive = below - above Total of this row Total.003*Pop..997*Pop.100,000

19. has CC~ have CCTotal tests positive = Pr(e|h) * [Pr(h)*Pop. ] = Pr(e|~h) * [Pr(~h)*Po p.] Total of this row ~ test positive = below - above Total of this row Total30099,700100,000

20. has CC~ have CCTotal tests positive = True Positive Rate (.9) * 300 = Pr(e|~h) * [Pr(~h)*Po p.] Total of this row ~ test positive = below - above Total of this row Total30099,700100,000

21. has CC~ have CCTotal tests positive 270= Pr(e|~h) * [Pr(~h)*Po p.] Total of this row ~ test positive = below - above Total of this row Total30099,700100,000

22. has CC~ have CCTotal tests positive 270= Pr(e|~h) * [Pr(~h)*Po p.] Total of this row ~ test positive 30= below - above Total of this row Total30099,700100,000

23. has CC~ have CCTotal tests positive 270= False positive rate (.03) * 99,700 Total of this row ~ test positive 30= below - above Total of this row Total30099,700100,000

24. has CC~ have CCTotal tests positive 2702,991Total of this row ~ test positive 30= below - above Total of this row Total30099,700100,000

25. has CC~ have CCTotal tests positive 2702,991Total of this row ~ test positive 3096,709Total of this row Total30099,700100,000

26. has CC~ have CCTotal tests positive 2702,9913,261 ~ test positive 3096,70996,739 Total30099,700100,000

27. has CC~ have CCTotal tests positive 270 (true positive) 2,991 (false positive) 3,261 ~ test positive 30 (false negative) 96,709 (true negative) 96,739 Total30099,700100,000

28. has CC~ have CCTotal tests positive 270 (true positive) 2,991 (false positive) 3,261 Wendy’s Chances given that she tests positive are the true positives divided by the number of total tests. That is, 270/3261, which is.083 (8.3%). Those who misestimate that probability forget that colon cancer is rarer than a false positive on a test.

29.  Note that testing positive (given the test accuracy specified) raises one’s chances of having the condition from.003(the base rate) to.083.  If we use.083 as the new base rate, those who again test positive then have a 73.1% chance of having the condition.  A third positive test (with.731 as the new base rate) raises the chance of having the condition to 98.8%