FOURSTATE ELECTRONIC RANDOM NUMBER GENERATOR
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January 16, 2003
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FIGURE 11 Fourstate electronic random number generator used
in this experiment. An incorrect choice of target
is indicated. Two of the five "encouragement lights"
at the top of the machine are illuminated. The
printer to the right of the machine records data on
fanfold paper tape.
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FourState Electronic Random Number Generator
This study provided an opportunity to determine whether the re
mote sensing capability could be extended to the perception of the
internal state of a piece of electronic equipment. For this purpose,
an automated experiment designed around a fourstate electronic random
number generator was initiated. The solidstate machine has no moving
parts and provides no sensory cue to the user as to its target genera
tion.
In order to determine unambiguously whether a result was meaning
ful, the following strategy was used. First, the randomness of the
machine was verified by over 10,000 preexperiment trials (details
given below). Second, the subjects interacted with the machine to
generate the data~ Third, for any subject whose score was significant,
the statistics of the machine during the successful experiment were
tabulated to insure that the machine had not departed from randomness
in the period in which a significant result was obtained. Fourth, a
subject generating a good score was asked to repeat the entire experi
ment after a onemonth lag period. Finally, the entire data analysis
was carried out by an independent statistics group at SRI. C Oz.
The machine configuration provides as a target one of four art
slides chosen randomly (p = 1/4) by an electronic random generator.
The generator does not indicate its choice until the subject indicates
his choice to the machine by pressing a button (see Figure 11). (The
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machine has four stable internal states. A 1.0NIIIz squarewave
oscillator sends pulses to an electronic "scaleoffour" counter which
passes through each of its four states 250,000 tifies per second. The
state of the counter is determined by the length of time the oscillator
has run, that is, the time between subject choices.) As soon as the
subject indicates his choice, the target slide is illuminated to provide
visual and auditory (bell if correct) feedback as to the correctness
or incorrectness of his choice.. Until that time, both subject and
experimenter remain ignorant of the machine's choice, so the experiment
is of the doubleblind type. Five legends at the top of the machine
face are illuminated one at a time with increasing correct choices (6,
8, 10, ...) to provide additional reinforcement. The machine choice,
subject choice, cumulative trial number, and cumulative hit number are
recorded automatically on a printer. Following trial number 25, the
machine must be reset manually by depressing a RESET button.
A methodological feature of the machine is that the choice of a
target is not forced. That is, a subject may press a PASS button when
he wishes not to guess, in which case the machine indicates what its
choice was. The machine thus scores neither a hit nor a trial and then
goes on to make its next selection. Thus, the subject does not have to
guess at targets when he does not feel that he has an idea as to which
to choose.
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Under the null hypothesis of random binomial choices with probability
1/4 and no learning, the probability of observing :~,.k successes in n trials
is approximated by the probability of a normal distribution value,
(k 2~/ 3u/16
PreExperiment Randomness Tests
The design objective was to build a fourstate machine, with each
state equally likely to occur on each trial, independent of the past
sequence of states. If the machine meets this objective, it should not
be possible to devise a rule for future play that significantly differs
from chance. A simple example of such a rule would be to select the
machine state observed in the preceding trial; if this strategy were
to produce scores significantly above chance (25 percent hits), we
would reject the hypothesis of randomness of the machine under test.
Before experimentation machines purchased from Aquarius Electronics,
Albion, California, were extensively tested for randomness. Data were
analyzed on a CDC6400 computer, and the machine finally selected for
use met established criteria for randomness.
In developing randomness tests, we are. guided in part by a knowledge
of the machine logic. When one of the four choice keys or the pass key
is depressed, the current machine state is displayed; then a brief time
after release of the key, a new machine state is established (but not
shown to the subject) by sampling the instantaneous state of a high
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speed fourstate electronic counter. For the machine to be random,
the times of dwell of the counter in each of the four states must be
precisely equal; otherwise, the distribution of outcomes will be biased.
The first randomness test is thus based on tallying the number of
occurrences of each of the four states. This test should detect a
stable bias, yet may miss a drifting bias. To test for this second
possibility we also tally the distribution of outcomes in each group
of 100 trials, then compute a likelihood ratio test statistic (see
below) for each group. Under the null hypothesis of equal likelihood
of the four states, these statistic values are distributed approxi
mately as chisquare with three degrees of freedom and their sum for
m groups distributed approximately as chisquare with three m degrees
of freedom. This test may also detect stable bias, but is not as
powerful for this purpose as the first test. Variable bias of still
a shorter period, if substantial, can be tested for by tallying the
frequency with which the previous machine state is repeated; an overall
repeat ratio ("all") significantly above 0.25 is indicative of such bias.
If for any reason the machine were to fail to sample the counter
to establish a new state, th previous machine state would be repeated.
To test for this possibility, we tally the number of repeats following
the depression of each key. A repeat ratio significantly greater than
0.25 should be considered a danger signal.
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We also tally the initial machine states following reset and the
transitions between states. In each case, the number of occurrences of
each of the four possible outcomes should be approximately equal. When
repeats are deleted from the sequence of trials ("nondiagonal transi
tions"), the four states should also be approximately equal in frequency.
In testing the null hypothesis of four equally likely outcomes of
a trial, a likelihood ratio test is used. The statistic
n ~n'
i=1
under the null hypothesis is distributed approximately as chisquare
with three degrees of freedom, with rejection for large values of this
statistic.* The computer program used in testing randomness includes
a subroutine for computing the probability of a chisquare value as
large or larger than that observed.
In testing the null hypothesis that the probability of a.repeat
is 0.25, the binomial probability of obtaining the observed number K
or more repeats in N trials is computed. For K greater than 1000, a
normal distribution approximation is computed, assuming the statistic
(; ` _N 0.251 3/16
to be approximately normal with mean.zero and standard deviation one.
*Alexander Mood, Introduction to the Theory of Statistics (McGraw Hill,
New York, 1950).
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Olt:
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The typical test pattern used was six passes followed by 25 choices
of one color, repeating this for each of the four colors. In this way
each of the five keys other than reset were given approximately equal
use. Typically, 2000 to 6000 trials were made in each sitting. In the
absence of any unusual results in the randomness tests, a minimum of
10,000 trials were made before using a machine with experimental subjects.
With 10,000 trials, the expected fraction of repeats is 0.25 with a
standard deviation of 3/200 = 0.00866.
A computer listing of the results of randomness tests is included
in Table 1. No significant departures from randomness were. observed.
Subject Data
Data was collected from subjects Si through S6. Each subject was
asked to complete 100 25trial runs (i.e_., a total of 2500 trials each).
The results are tabulated in Table 2. .(One subject, S3, declined to
complete the 2500trial run, indicating a lack of rapport with the
machine and, hence, a lack of motivation for the task.)
subjects, only one (S2) scored significantly above chance. For the
2500 trials that subject averaged 29.36 hits/100 trials rather than the
expected 25/100, a result whose a priori probability under the null
hypothesis is p = 3x10 7 . His scores are plotted in Figure 12.
The statistics of the machine during the successful run of subject
S2 were tabulated for the entire 3488 machine transitions (2500 choices,
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Table 1
PREEXPERIMENT RANDONWESS TESTS
Buttons Number
of
Yellow Green Blue i Red Trials
Transitions Y
G
From To '' B
R
All states
Nondiagonal
transitions
Diagonal
transitions
Diagonal
transitions
as a function
of key press
728
777
776
787
2340
Yellow
Green
Blue
764
784
796
852
765
773
810
803
Randomness in groups of 100 trials:
Chisq. = 299.6141 D.F. = 345
790
863
773
805
Repeats
3047
3197
3155
3247
0.2541
0.2446
0.2557
0.2433
0.2323
0.2473
2.573
6.745
1.158'
2.877
5.667
Binom.
Prob.
0.46
0.08
0.76
0.41
Bionomial
Prob.
0.313
0.748
0.250
0.793
0.953
0.763
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TABLE 2
FOURSTATE ELECTRONIC RANDOM NUMBER GENERATOR
MEAN SCORE/100 TRIALS
BINOMIAL
SUBJECT
OVER 2500 TRIALS
PROBABILITY
Si
25.76
0.22
S2
29.36
7
3 x 10
S3
24.67 (750 trials)
0.60
S4
25.76
0.22
S5
25.20
0.42
S6
25.40
0.33
4
S7
27.88
4.8 x 10
(replication)
5
All trials
26.47
1.1 x 10
(15750 trials)
r
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MIDEXPERIMENT RANDOMNESS TESTS
.BUTTONS
Number
f
Binom.
Yellow
Green
Blue
Red
o
Trials
ChiSq.
Prob.
Initial States
24
29
23
24
100
0.880
>0.80
Transitions Y
_204
199
199
216
818
0.944
>0.80
To G
1 0 192
223
222
207
844
3.043
>0.30
From B
212
207
226
222
867
1.064
>0.70
R
t" 209
207
222
221
859
0.860'
>0.80
1.
All States
841
865
892
890
3488
1.988
>0.50
Nondiagonal
_

Transitions
613
613
643
645
2514
1.535
>0.50
Diagonal '
Transitions
204
223
226
221
874
1.341
>0.70
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50
RUN NU\U3ER  100 Trials/Stun
p= per trial
FIGURE 12 DATA SUMMARY FOR SUBJECT 2
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