Complete discussion of basic probability from last time

Applied Probability Problems - see handout

Conditional Probability - we will return to this after the quiz

Class Exercises - some additional examples of Simpson's Paradox

1. Suppose that the manager of the Cleveland Guardians is faced with a difficult situation in the last inning of a baseball game later this evening and he asks for your input. Here is the situation. He has two right-handed batters, say Moe and Joe, available to pinch hit in the last inning to try to win the game. However, he is not sure which player to use. The performance of the players during the 1995 season is shown in the table below.

Player Pitcher Hits At Bats
Joe Right 40 100
Left 80 400
Moe Right 120 400
Left 10 100

a. Construct a two-way table of player (Joe or Moe) versus outcome (hit or no hit).

b. Compute the overall batting average (hits divided by total number of times at bat) for each of the players.

c. For each of the two types of pitching, construct a two-way table of player by outcome (hit or no hit).

d. Use the tables in part (c) to compute the batting averages of the two players for each type of pitcher.

e. Based on the calculations in (b) and (d), who would you recommend? Explain your recommendation in language that the manager will understand.

2.  The University of California at Berkeley was charged with having discriminated against women in the graduate admissions process for the fall quarter of 1973. The table below identifies the number of acceptances and denials for both male and female applicants in each of the six largest graduate programs at the institution at that time:
Men Accepted Men Denied Women Accepted Women Denied
Program A 511 314 89 19
Program B 352 208 17 8
Program C 120 205 202 391
Program D 137 270 132 243
Program E 53 138 95 298
Program F 22 351 24 317
Total 1195 1486 559 1276

(a) Start by ignoring the program distinction, collapsing the data into a two-way table of gender by admission status. To do this, find the total number of men accepted and denied and the total number of women accepted and denied.
Admitted Denied Total

(b) Consider for the moment just the men applicants. Of the men who applied to one of these programs, what proportion were admitted? Now consider the women applicants; what proportion of them were admitted? Do these proportions seem to support the claim that men were given preferential treatment in admissions decisions?

(c) To try to isolate the program or programs responsible for the mistreatment of women applicants, calculate the proportion of men and the proportion of women within each program who were admitted. Record your results in a table like the one below.
Proportion of Men Admitted Proportion of Women Admitted
Program A
Program B
Program C
Program D
Program E
Program F

(d) Does it seem as if any program is responsible for the large discrepancy between men and women in the overall proportions admitted?

3. The influence of race on imposition of the death penalty for murder has been much studied and contested in the courts.  The following three-way table classifies 326 cases in which the defendant was convicted of murder.  The three variables are the defendant's race, the victim's race, and whether the defendant was sentenced to death.
White Defendant
Death Penalty No Death Penalty
White Victim 19 132
Black Victim 0 9

Black Defendant
Death Penalty No Death Penalty
White Victim 11 52
Black Victim 6 97

a. Construct a two-way table of defendant's race by death penalty.
b. Show that Simpon's paradox (see Section 2.5) holds: A higher percent of white defendants are sentenced to death overall, but for both black and white victims a higher percentage of black defendants are sentenced to death.
c.  Explain why the paradox holds in language that your roommate would understand.

Reminder: Quiz #1 will be given in class on Friday. You may use our textbook, your notes, our R scripts, and our web page during the quiz. The quiz will cover the material in Chapters 1 and 2 of our textbook and any additional material that we have covered in class.

Please read sections 3.4 and 3.5 for class on Monday