Illogical Inductive Conversions

Transcription

1 Chapter 11, Part 7 Illogical Inductive Conversions One more caution regarding inductive reasoning:

2 Stats show that of drivers involved in fatal crashes, 40% are under 26 years old. Therefore, if you re a driver under 26 years old, your chances of being involved in a fatal crash are 40%??? 2

3 Stats show that, of drivers involved in fatal crashes, 40% are under 26 years old. Therefore, if you are a driver under 26 years old, your chances of being involved in a fatal crash are 40%. Good argument? A.Yes B.No 3

4 NO, and it is important to see why. 4

5 Percent of drivers involved in fatal crashes who are under 26: 40% Percent of drivers not involved in fatal crashes who are under 26: 20%. Percent of drivers involved in fatal crashes: 1%. So out of every 1000 drivers, 10 (1%) are involved in fatal crashes; 990 aren t. Of 10 who are, 40% are under 26 = 4 Of 990 who aren t, 20% are under 26 = 198 Total drivers under 26: of the 202 are in fatal crashes: 2% 5

6 Stats show that 40% of drivers involved in fatal crashes are under 26 years old. Therefore, if you are a driver under 26 years old, your chances of being involved in a fatal crash are 40%. Stats show that 40% of drivers involved in fatal crashes are under 26 years old. Therefore, if you are a driver 26 years old, your chances of being involved in a fatal crash are 2%. 6

7 Moral: n% of As are Bs. Therefore, n% of Bs are As. You cannot get to that conclusion without additional information! 7

8 Here we needed to know: the percent of drivers NOT involved in fatal crashes who are under 26 AND the percent of drivers involved in fatal crashes. 8

9 This is complicated. All you need remember is this: 9

10 n% of As are Bs. n% of Bs are As. You cannot get to that conclusion without additional information! 10

11 This is just common sense. 90 % of terrorists are Muslims. 90 % of Muslims are terrorists. 11

12 Likewise: Most Terrorists are Muslims. Most Muslims are terrorists. 12

13 Likewise: Most Democrats are liberals. Most liberals are Democrats. 13

14 Let s apply this to one more important real-life type example... the kind of thing many of us encounter at some point. 14

15 George has tested positive for bladder cancer. The test is 90% accurate (90% of those who have bladder cancer test positive, and 10% of those who don t have it, test positive). How probable is it that George has bladder cancer? A. Extremely probable B. Very probable C. Probable D. About 50/50 E. Improbable Let s see! 15

16 George has tested positive for bladder cancer. The test is 90% accurate (90% of those who have bladder cancer test positive, and 10% of those who don t have it, test positive). How probable is it that George has bladder cancer? Well, just because 90% of people with bladder cancer test positive, it does NOT follow that 90% of those who test positive have bladder cancer! 16

17 Suppose the rate for bladder cancer is 1%. Then, for every 1000 people: About 10 will have bladder cancer. Who will probably test positive. And 990 people will not have bladder cancer. And 10% of those will test positive (99). So: a total of 109 will test positive. Of the 109 who test positive, 10 will have bladder cancer: That s 9% Given this information, the probability George has bladder cancer if he tests positive is really just 9% not 90%. 17

18 A good way to avoid mistakes like these, which are easy to make, is to remember: n% As are Bs. n% Bs are As? No! 18