An Introduction to Data Analysis & Presentation Prof. Timothy Shortell, Sociology, Brooklyn College Probability Up to now, we have been concerned with describing data. We are going to begin studying the process of statistical decision making, or
Decision making is based on probability. When we engage in hypothesis testing, we are balancing the possibility of making the correct decision with the possibility of making an incorrect one. We need to understand some of the principles of probability in order to assess when we have the right balance. We can define probability as Suppose that there are 6 candidates in the Republican primary for President. Two are women. One is an African American man. Consider this: Suppose that there are 20 soloists in the Brooklyn College Symphony Orchestra. 12 play a string instrument, 5 play a horn, 2 a wind instrument, and one is a purcussionist. What is the probability that the next soloist to walk onstage will perform a Beethoven violin concerto?
Suppose that there are 15 faculty at Brooklyn College in Sociology. There are 21 faculty in Political Science and 28 in History, but only 6 in Anthropology. There are 300 faculty in other departments. Assuming that these four departments make up the Social Science Division at the college, what is the probability that the next Provost will come from the Social Sciences? What we are asking here is: what is the probability that the next Provost will come from Sociology Another example: Suppose that 200 researchers have applied for a grant from the National Science Foundation. 134 applicants come from public universities. 46 come from public liberal arts colleges. 15 come from private universities. The remaining 5 come from private research institutes. What is the probability that the grant will go to someone at a public school? At a university? At any institution of higher education?
Suppose that your statistics instructor has 15 coats. 7 are black, 4 red, 3 blue and one green. What is the probability that the instructor will wear a black coat on Tuesday and a red one on Wednesday? A black coat both days? A blue coat and then a red one? Here's another one: Suppose that downtown Urbanville has 16 avenues, 13 streets, 7 boulevards and 4 parkways. What is the probability that a traffic jam will occur on an avenue and a street? An avenue and a parkway? On all the parkways?
Another common problem is the One interesting instance of faulty logic about probabilities involves the conjunction of any two events that seem to go together. If you look closely at the rules for probability we've just discussed, you can see that the probability of A and B occuring is always smaller than the probability of A or B alone (as long as the events A and B have probabilities less than one, that is, are not certainties). Thus if the probability of A is 0.5 and the probability of B is 0.25, then, applying the multiplication rule, we know that the probability of A and B occuring is 0.125. which is less than either A or B. Consider this example: Which of the two outcomes is more likely?
If you think about it as an application of the multiplication rule (A and B) then you realize that the conjunction has to be less likely. (Taken from Let's put some numbers to the problem. (It doesn't matter that the probabilities are not empirically accurate.) Let's say that the probability of having had a heart attack is 0.10 and the probability of being over 55 years of age is 0.33. Then the probability of both having had a heart attack and being older than 55 is 0.03, which is less than either single probability alone.
All materials on this site are copyright © 2001, by Professor Timothy Shortell, except those retained by their original owner. No infringement is intended or implied. All rights reserved. Please let me know if you link to this site or use these materials. |