Chapter 6
Bernoulli Trials Revisited
6.1
Is p Constant?
Recall that there are three assumptions to BT. The first, that each trial yields a dichotomy, is easy
to verify or refute. The other two assumptions can neither be verified nor refuted with certainty.
In this Section we present methods for investigating the second assumption, that the probability of
success remains constant. Our methods will consist of informal descriptive techniques and formal
tests of hypotheses. Both of these categories of methods are very useful to a scientist.
I begin with an extended example of data I collected circa 1990 on Tetris.
Tetris is a video game (I fear that this expression—video game—is hopelessly dated; perhaps
you can give me a better name to use for the next version of these notes!) that can be played on a
variety of ‘systems.’ If you have never played Tetris, then the following description of the game
may be difficult to follow. Tetris rewards spatial reasoning and, not surprisingly, lightning reflexes.
Essentially, one of seven possible geometrical shapes of four blocks ‘falls’ from the top of the
screen. The player translates and/or rotates the falling shape in an attempt to complete, with no
gaps, a horizontal row of blocks. Each completed row of blocks disappears from the screen and
blocks above, if any, drop down into the vacated space to allow room for more falling shapes. A
player’s score equals the number of rows completed before the screen overflows, which means that
the current game is over. After every ten completed lines the speed of the falling shapes increases
making the game more difficult.
Each drop of a shape by the computer can be viewed as a trial. Each trial has seven possible
outcomes. For now, the shapes are divided into two types: a straight row (my favorite, labeled a
success) and any other shape (a failure).
I observed 1,872 trials during eight games of Tetris on my son’s Nintendo System. The methods
I present below are based on taking the total number of trials, 1,872 in the current case, and dividing
them into two or more segments. The choice of segments is a matter of taste and judgment and
unless I have a good reason to do otherwise, I advocate dividing the data into segments of equal
or nearly equal length, and I usually advocate having two segments of trials. For these Tetris data,
however, there is a structure that I want my analysis to reflect, namely, every time a game ends
the system is ‘reset’ before the next game. (There was literally a button that said ‘Reset.’) Thus,
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Table 6.1: A Comparison of Eight Games of Tetris.
Frequencies
Row Proportions
Game
S
F
Total
S
F
Total
First
22
171
193
0.114
0.886
1.000
Second
33
185
218
0.151
0.849
1.000
Third
36
215
251
0.143
0.857
1.000
Fourth
25
206
231
0.108
0.892
1.000
Fifth
30
198
228
0.132
0.868
1.000
Sixth
42
220
262
0.160
0.840
1.000
Seventh
33
215
248
0.133
0.867
1.000
Eighth
33
208
241
0.137
0.863
1.000
Total
254
1618
1872
0.136
0.864
1.000
Figure 6.1: Plot of Proportion of Success for Eight Games of Tetris.
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 Fall '09
 ProfessorWardrop
 Statistics, Bernoulli, Null hypothesis, Statistical hypothesis testing, ... ..., total

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