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THE APPLICATION OF TWO-ARMED BANDIT 
STRATEGIES TO CLINICAL TRIALS 
by 
Donald A. Berry 
University of Minnesota 
Technical Report No. g65 
May 1976 
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THE APPLICATION OF TWO-ARMED BANDIT 
STRATEGms TO CLINICAL TRIALS 
by 
Donald A. Berry* 
University of Minnesota 
Technical Report No. 265 
ABSTRACT 
A procedure which maximizes the expected number of successes in a clinical 
trial involving two treatments can usually be found only by backward induction. 
Not only is it difficult to find an optimal procedure but, once found, it is 
difficult to describe and cumbersome to communicate. A procedure is suggested 
which depends on the information present concerning the treatments, is easy to 
calculate, and approximates an optimal procedure quite well. The procedure is 
applicable to trials for which the number of patients is unknown as well as 
those of fixed duration. It can be modified to apply as well in trials in which 
there is a delay between treatment application and response. 
I 
*!'his work; was supported by Public Health Service Research Grant No. GM 22234-01 
.from the National Institutes of Health. Computations were performed by Larry M. 
Pearson while supported by a Research Assistantship from the Graduate School of 
the University of Minnesota. 
--
THE APPLICATION OF TWO-ARMED BANDIT 
I 
STRATEGIES TO CLINICAL TRIALS 
by 
Donald A. Berry 
1. Introduction. 
Suppose that two treatments are available for use in a clinical trial. 
Further suppose that the response to treatment is either positive, a success, 
or negative and that the patients can be treated one at a time, with each 
patient's response available before the next patient is to be treated. Since 
this assumption is seldom realistic, modifications will be considered in 
Section 6. 
The number of patients in the trial is N, which may be considered 
fixed or random; for convenience the consideration of the latter possibility 
will be delayed until Section 7. The main purpose of the trial is to treat 
these N patients as effectively as possible. An inevitable and obviously 
beneficial result of any trial is the acquisition of information concerning 
the treatments. An effect of this accumulating information is that it allows 
for better treatment of patients appearing late in the trial than those appear-
ing early. The treatment allocation problem for early phases of the trial is 
then governed by two possibly conflicting desirata: (1) A patient should be 
given the treatment which is apparently superior. (2) A patient should be 
given the treatment about which more is likely to be learned. 
Suppose that the probability of success using Treatment 1 is and 
using Treatment 2 is If and are known precisely then there is 
no information to be collected during the course of the trial, and the best 
plan for the trial is to allocate all patients to the treatment which has the 
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larger pi' and for that allocation the expected number of successes is 
N max{p1 , p2 }. If the pi are unknown then there are at least two approaches 
to the problem. One, the classical approach, suggests a class of allocation 
schemes or sampling procedures and finds the expected number of successes, for 
example, using each procedure as a function of (p1 , p2 ). Of course, no one 
procedure is better than all other procedures uniformly in (p1 , p2 ), so, in a 
sense, the procedure which one should use depends on knowing p1 and 
But if p1 and p2 are known the best procedure is the trivial one described 
above! Some of these procedures will be discussed in Section 2. Another, the 
Bayesian approach, asks that current information concerning (p1 , p2 ) be quan-
tified in the form of a probability distribution. It is this approach which 
will be used in most of this paper. A distinct advantage of the approach is 
that accumulating information can be handled in a unified way: Bayes' theo-
rem is used to modify the probability distribution on (p1 , p2). The effect-
iveness of a procedure can then be averaged over (p1 , p2 ) and, hopefully, 
an optimal procedure found. 
If one of the is known, say p1 , and p2 has probability measure 
v(p2 ), then the procedure selection problem' is called a "one-armed bandit;" 
cf. Bradt, Karlin, and Johnson (1956). There are two frequently occurring 
clinical trials for which this problem is appropriate: (1) Treatment 1 is a 
standard treatment and has been used many times, and Treatment 2 is experimental; 
(2) Treatment 1 is a placebo and Treatment 2 experimental. If Ep2 > p1 , where 
expectation E is with respect to v, then using Treatment 2 not only has more 
information value but also has a greater probability of success (this being the 
expected value of p2 ) associated with it; it seems clear that Treatment 2 
should be used, at least until such time as Ep2 < p1 • It may not be as clear 
that Treatment 2 may be optimal even though Ep2 < p1 • 
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In fact, Treatment 2 may be used optimally even though it yields only failures 
for an arbitrarily long time, depending on N, and provided v(p2 > p1) > O. 
A variant of this problem which avoids the latter difficulty discounts success-
ful treatment of future patients, so that the present patient is worth 1 
while the next is worth y and the following y2 , etc., for O ~ y < 1; cf. 
Section 7, Bellman (1956), and Berry (1976). 
An intuitively appealing characteristic of optimal procedures for the 
one-armed bandit is that the initial patients constitute an information 
gathering stage, which may be empty or may exhaust the trial, during which 
the experimental Treatment 2 is used, and then the standard Treatment 1 is 
used for the remainder of the trial. Whether information gathering is initia-
ted depends on v; for example, a necessary condition is v(p2 > p1 ) > O. 
The length of this stage depends on the information accumulat~d concerning p2 
'and, to a decreasing extent, the initial distribution V of 
and are both unknown and subject to a probability measure 
then the problem is a "two-armed bandit." Unlike the case in 
which p1 is known, sampling can switch from one treatment to the other any 
number of times using an optimal procedure. With one important exception 
to be mentioned immediately, an optimal procedure can only be found using a back-
ward induction requiring on the order of N3 storage locations and is, in 
general, extremely difficult to implement since it must specify a treatment to 
use in each of 4N different situations. If however, ~(a,~)+µ(~, a)= 1 
for any fixed pair {a,~), then an optimal procedure is especially easy: 
Treatment 1 is used whenever the current probability that pi= max{a, ~} is 
at least 1/2. {This result is due to Feldman (1962), and was generalized by 
Fabius and van Zwet (1970), Berry (1972), and Kelley (1974).) The only calcula-
tion that is necessary in implementing this procedure is that of the posterior 
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probability that (p1 , p2 ) = (a, S). Such procedures are discussed further 
in Section 3 of this paper. 
A method is suggested in Section 4 for approximating µ with such a 
two-point measure. The loss in effectiveness in using the procedure that is 
optimal for this approximating measure is examined in Section 5. 
2. The classical approach • 
The two-armed bandit problem was considered by Robbins (1952), who 
suggested the so-called "play-the-winner rule:" randomly select one of the 
treatments to be used on the first patient and henceforth use the same treat-
ment on the next patient whenever the response is positive and the other 
treatment whenever the response is negative. Though this procedure is not 
optimal in general for any u, it behaves well despite its simplicity. If 
µ, is such that P1 and p2 are independent, then Berry (1972) shows that 
an optimal procedure always "stays on a winner," but sometimes switches and 
sometimes stays on a loser. Bradt, Karlin, and Johnson (1956) give an example 
in which pl and P2 are dependent and for which the unique optimal pro-
cedure switches on a winner and stays on a loser! 
Robbins' approach has been generalized to allow the performance of the 
treatments on the previous k patients to dictate the treatment used on the 
next patient; this is the so-called "finite memory" approach (for references 
see Smith a~d Pyke (1965)). 
There is a very large "selection and ranking" literature which examines 
different saq,ling procedures. Many of the better procedures use Robbins' 
play-the-winner rule in the course of the trial, but effectively treating 
the patients in the trial is regarded as being of secondary importance - of 
primary importance is determining which treatment has the larger p at the 
end of the trial. There are many papers which use this approach referred to 
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in Simon et al. (1975). 
A procedure which is designed to treat successfully as many patients as 
possible in the course of the trial was suggested by Zelen (1969). Zelen uses 
·Robbins' play-the-winner rule for the first n patients in the trial and then 
uses the apparently superior treatment for the remaining N-n patients. Zelen 
finds that n ~ N/3 yields nearly the maximal expected proportion of successes 
for the entire trial. In one example he supposes that N = 100, p1 = .75, and 
p2 = .25. The maximal expected proportion of successes using his procedure is 
.726, which corresponds to n = 14, while the proportion is .709 for 
n = 33. 
To compare Zelen's procedure with the optimal procedure, which is Feldman's, 
fix µ(.75, .25) = µ(.25, .75) = l/2. Feldman's procedure is especially simple 
when µ{a, 1-a) = µ(1-a, a)= l/2; for the current patient it requires using 
the treatment for which the current difference between number of successes and 
number of failures is greater (cf. (3.2)), and uses either when these differences 
are the same. The total expected propQrtion of successes when N = 100 using 
Feldman's procedure is .740. Now this number is, as usual, an average with 
respect to µ, which is a distinctly Bayesian concept. However, in view of the 
special nature of µ, this number is constant over its entire support so this 
··number has a non-Bayesian interpretation as well and is directly comparable with 
.709 and .726 of Zelen's procedures - of course, no procedure can do better 
than Feldman's. Incidentally, .740 compares favorably with .75, the expected 
proportlon attainable if the better treatment were known a priori. Put in 
these terms, it is clear that Feldman's procedure uses the better treatment 
(pi= 3/4) an average of 98 of the 100 trials! 
At least two other papers of importance should be mentioned hereo 
Anscombe (1963) and Colton (1963) independently proposed a two-stage procedure 
similar to Zelen's except that both treatments are used the same number of times 
} 
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in the initial, information gathering stage. This was in turn generalized 
by Cornfeld et al. (1969) to allow for an arbitrary proportion of patients 
on.Treatment 1 in the initial stage. 
3. Myopic procedures. 
There is a slightly different way of viewing Feldman's procedure that 
can help put it in perspective. If N = 1 then an optimal procedure is to 
use Treatment 1 if Ep1 ~ Ep2 and Treatment 2 otherwis,e, where expectation 
is with respect to µ. If µ(a, a)= r = 1-µ(a, a) and a> a then 
Epl - Ep2 = ra + (1-r)a - [(1-r)a + ra] = 2(a-a)(r-½) has the same sign as 
r - ½. Therefore, Feldman's procedure behaves as though each patient is the 
only patient in the trial; that is, the procedure is "myopic." 
At any stage of the trial let s 1 , f 1 , s 2 , f 2 be, respectively, the 
numbers of successes and failures on Treatments 1 and 2. The probability 
( pos teri or probability) that p1 = a is then 
f) = l + 1-rc~) 1 2(1-a) 1 2 
[ 
s -s f -f ]-l 
2 r ~ l-G' 
which is > ½ if 
In case a= 1-a this is equivalent to 
and further, if r - .J.. 
- 2 it is equivalent to 
as was stated in Section 2Q 
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While the optimal procedure for general µ is not known, it has been 
partially characterized in Berry (1972) when p1 and p2 are independent. 
Th~ myopic procedure for (a, S) arbitrary has at least one characteristic 
in connnon with the optimal procedure: it "stays on a winner." A success on 
Treatment 1 increases the left-hand side of (3.1) while a success on Treat-
ment 2 decreases it. Also, a failure may leave inequality (3.1) unchanged 
or it may reverse it; the difference between a myopic procedure and an 
_optimal procedure is that one may switch on a failure when the other does 
not. 
4. The class of procedures B(µ). 
The procedure B{a{µ), S{µ)), abbreviated B(µ), is defined to be 
Feldman's procedure with a= a(~), S = S{µ), and r = r{µ); that is, 
Treatment 1 is used whenever inequality (3.l) holds and Treatment 2 is used 
whenever the inequality in (3.l) is reversed. When the inequality in (3.l) 
is replaced with equality then either treatment can be used following 
Feldman's procedure; B(µ) may strictly prefer one of the treatments depend-
ing on µ, as will be described shortly. 
There is no obviously best way to define a{µ), S(µ), and r(µ). While 
there are many candidates, the definition proposed here is natural and, as 
will be seen, gives excellent results. 
Make the simplifying assumption that µ is such that µ{p1=p2 ) = O. 
This is without loss of generality since if µ{p1=p2 ) > 0 and µ is changed 
to 
= lµ(pl, P2)/µ{p1=P2), 
o, 
for 
for 
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then the optimal procedure is unchanged. The point is that when p1 = p2 
the treatments are the same and on the line p1 = p2 every procedure has 
the same effectiveness. 
For arbitrary µ define 
r(µ) =µ(pl> p2) = Jdµ(pl, p2). 
pl>p2 
Assume r(µ) ~ ½; if it.is not then reverse the names Treatment 1 and 
Treatment 2. This means that Treatment 1 is used initially when following 
B(µ), except possibly when r(µ) = ½• Consider the measure 
µ*(pl, P2) 
and define 
(4.1) r1(µ) = E*P1, 
= 
µ(pl, p2)/r(µ), 
0, 
a(µ)= E*P2 , 
for Pl> P2 
for Pl :'.S P2 
where E* denotes expectation with respect to µ*. So defined, a(µ) and 
~(µ} ·are easy to calculate and are simply the conditional expectations of 
Pi and p2 given pl> p2 • An alternative definition that comes to mind 
immediately corresponds to conditioning on p1 < p2 instead; however, compar-
isons suggest that conditioning on the triangle with larger probability is 
usually preferable. 
Procedure B{µ) is myopic and is not optimal in general, no matter how 
a(µ), ~(µ), and r(µ) are defined. Kelley (1974) explores conditions 
under which every optimal procedure is myopic. B(µ) is compared with an 
optimal procedure in Section 5 for various µ which have densities in the 
plane: 
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(4.2) 0 < pi < 1, i = 1,2, 
and v1 and v2 are beta densities: 
(4.3) 
acl bi-1 
v1 (x) = kix (1-x) , O<x<l. 
The v. are densities for a.> 0 and b. > o. Under this assumption 
L L 1. 
and P2 and independent and Ep. = a./(ai + b.), 1. 1. l. i = 1,2. Also, where 
and fi are defined in· Section 3, for i = 1, 2, 
Table l gives the values of ~(µ,}, a(µ,}, and r(µ,) for various v1 and 
v2 as given by the pairs (a1 , b1) and (a2 , b2 ). 
P1 
s. 
l. 
The initial treatment to be used following B(~} is as yet not specified 
when r(µ,) =½,or at any later stage when equality replaces the inequality in 
(3. l). In such a case B(µ,) is defined to use Treatment i if "less is known" 
about pi than p., 
J 
j I: i. When µ, is of the form (4.2), this is taken to 
mean using Treatment i if ai +bi< aj + bj; when they are equal either 
treatment is used. When pl and p2 are not independent beta variables then 
the definition of "less is known" is le£ t open. While there are several 
reasonable definitions of the information contained in a probability distribution, 
these are avoided here because the matter is usually an insignificant one. 
Indeed, u may be such that it is impossible for the probability of (p1 > p2 ) 
ever to become exactly ,t. It can be important if a great deal is known about 
Treatment 1, say, and little about Treatment 2. In such a circumstance B(µ,} 
could be effectively modified by requiring the use of the Treatment 2 whenever 
r(µ,} is somewhat less than ½ - this is the character of the optimal pro-
cedure in the one-armed bandit, in which one treatment is precisely known. 
A limitation of B(µ,) is that there is no way of letting a two-point 
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measure reflect the possibility that "more is known" about one treatment 
than the other. 
5. Comparison of B(u) and optimal procedure. 
Consider first the special case in which µ, is the product of uniform 
measures on (0, 1), so that p1 and p2 are independent and, in terms 
of the notation of (4.2) and (4.3), a1 = b1 = a2 = b2 = 1. Then, by 
synnnetry, r{µ,} = ½, and 
a{µ,} 1 Pl 2/3, = 2J J p1dp2dpl = 0 0 
a{µ,) = 2J1l1 p2dp2dp1 = l/3. 0 0 
Since r(µ,} = ½ and a(µ,)= 1-a(µ,), procedure B(µ,} uses Treatment 1 
whenever (3.2) holds and Treatment 2 whenever the inequality is reversed 
in (3.2); whenever (3.2) holds with equality then Treatment 1 is used if 
s1 + f 1 ~ s2 + f2 and Treatment 2 otherwise. B(µ,) is an optimal proce-
dure for values of N < 5 and imitates an optimal procedure much of the time 
-for larger values of N. This is seen in Table 2 which gives the expected 
proportion of success for B(µ,) and for optimal procedures when 
N = 1(1)10(5)40(20)100. It is to be stressed that these expectations for 
B(~) are calculated with respect to µ, and~ with respect to the indicated 
two-point measure; the clinician is acting as though µ, were a two-point mea-
sure but he knows the actual µ,. As N-+ 00 the limiting proportion for 
B(µ,) and optimal procedures is 
and the convergence is monotonic in both. 
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Table 3 gives the expected proportion of success using B(~) and 
using an optimal procedure when N = 25 for each of the pairs of beta 
densities considered in Table 4.1. Table 4 compares these procedures 
when N = 50. These numbers were computed by backward induction. In 
addition, the values of max(Ep1, Ep2) and of E max(p1, p2) are given 
for each pair. These two numbers give effective lower and upper bounds 
for all reasonable procedures and all N. It is readily seen that in 
quite a broad range of measures µ the procedure B(µ) compares favor-
ably with an optimal procedure. 
As indicated before, the manner in which r(µ), a(µ), S(µ) were de-
: fined ·in Section 4 is not necessarily "optimal" - the procedure B(u,) may 
not be the best myopic procedure. Calculations not reported here were ma.de 
for various triples (r, a, S) and various· u and N; they make it clear 
that when B(µ) is not optimal among myopic procedures it is nearly so. 
6. Delays in treatment response. 
In most clinical trials there is a nontrivial time delay be~een the 
administration of a treatment (as a drug) and a response to the treatment. 
During this delay other patients may have to be treated - possibly for 
considerations of time as well as of patient health. If this delay is so 
great that no~e of the patients will respond before the Nth patient is 
treated then the procedure is easy: simply assign all patients to the 
treatment with the larger probability of success, given by EP1 or Ep2 • 
As is evident here, and is true more generally, there is a decrease in the 
maximal expected proportion of success when there is response delay. 
Suppose that the delay is in terms of a fixed number, n, or patients, 
so that n + 1 patients are treated before the first patient's response be-
comes known. When u is such that µ(a, S) + µ-{S, a) = 1 then an induc-
tion on n shows that the myopic procedure is again optimal. In this case, 
of course, the current probability that p1 > p2 is used to determine which 
treatment to use, and since this probability does not change until the (n +l)st 
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·- patient is treated, at the least the first n + 1 patients are given the same 
treatment. Also, except for the first n patients one response becomes known 
immediately after each treatment and the information contained in a response is 
independent of the treatment used, so that the last N-n patients.!!'..!:. treated!!. 
though they constitute !. trial with !!2 respom e delay. If, as in the example in 
Section 3, ~(.75, .25) = µ(.25, .75) = ½, assume N = 200 and n = 100; then 
i 
the expected proportion of success over the trial is 
100 i 100 200<½(.75) + ~(.25)) + 200<-740) = .250 + .370 = .620, 
which compares with .5 if n ~ 199 and .745 if n = O. If n is considered 
to be random, and varying from one patient to another, then this result extends 
to show that the myopic procedure is again optimal. 
The determination of an optimal procedure for general ~ does not follow as 
easily from the case of no response delay. It is not in general true, for 
example, that the first n + 1 patients should be given the same treatment. The 
obvious generalization of B(µ) to this case, approximating µ with a two-point 
'measure and allocating accordingly, is not as efficient as when n = O. It seems 
reasonable to modify B(µ) so that the first n patients are allocated between 
the treatments so that the expected amount of information present after n re-
sponses become known will be approximately the same for both treatments. Another 
possibility, which could be combined with the first, is to assign r(µ,) of the 
first n patients to Treatment 1 and 1-r{µ) of them to Treatment 2. The re-
maining N-n patients can then be allocated following B(µ). The effectiveness 
of these procedures has not been considered here. The optimal procedures for the 
delayed response problem have never been systematically studied. If n is allowed 
to be random rather than fixed then it is not obvious how best to modify B(u) 
and comparisons become increasingly complicated. There are a variety of kinds 
of "randomness" to consider. The delay in response may be less for one treatment 
{which may not be known) or it may depend on whether or not· the response 
is positive. In such situations there is still a third consideration in 
using a treatment: the desire to use a treatment which will likely yield 
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,., information more quickly so that information can be c~s idered in the treatment 
of more patients. This desirata is inseparable from the other two that have been 
discussed previously - using a treatment with a large expected probability of 
• success and one which yields a large amount of information. 
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7. Unknown trial length. 
It is frequently the case that N is not precisely known, but is subject to 
a probability distribution. Assume that N is independent of (p1 , p2 ) and of 
the way in which the trial is conducted. If µ{a, S) + µ{S, a)= 1 then the 
optimal procedure, being myopic, is not affected. The procedure B(u,) does not 
depend on N though, of course, the optimal procedure does depend on N. Roughly 
speaking, an optimal procedure is more inclined to sacrifice iIIllllediate payoff for 
information that can be used later. B(µ) is offered here as a reasonable approxima-
tion to the optimal procedure for all values of N and, therefore, it is reason-
able for any distribution of N. In fact, since the one-point distributions are 
the extremes, the distribution of N for which B(µ) suffers most in comparison 
with the optimal procedure is one in which N is known. In this sense, not 
knowning N actually improves the appropriateness of B(~). 
These remarks hold as well if N is effectively infinite and a success 
. 1 
on the jth patient is discounted by the factor YJ- , where O :SY< 1. Then 
the ·expected discounted number of successes is 
plays the role of N(though not in every sense). 
bounded above by (1-y)-l which 
Posed thusly, this problem 
is similar to the one in which N is random, with a geometric distribution; 
there is a constant probability, 1-y, for each patient that the trial will 
terminate with that patient. 
8. Summary and conclusions·. 
A procedure which maximizes the expected number of successes in a clinical 
trial has been shown to be reasonably approximable with a procedure that is rela-
tively easy to calculate. This procedure, B(µ), is determined from µ, the 
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available information concerning the treatments, using elementary calculations, 
and without the need for backward induction. B(u) is actually optimal for 
certain µ and compares favorably with an optimal procedure in situations for 
which it is not optimal. The procedure is appropriate, possibly with modifica-
tion, when there is a delay between treatment application and response and 
when the length of the trial is not known. 
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REFERENC F.S 
Anscombe, F. J.(1963). Sequential medical trials, Journal of the American 
Statistical Association, -2.§, 365-383. 
Bellman,_R. E.(1956). A problem in the sequential design of experiments, 
Sankhya, 16, 221-229. 
Berry, D. A.(1972). A Bernoulli two-armed bandit, Annals of Mathematical 
Statistics, il, 871-897. 
Berry, Do A.(1976). One-armed bandits and related problems. Technical 
Report No. 267, School of Statistics, University of Minnesota. 
Bradt, R. N., Johnson, s. M., and Karlin, s.(1956). On sequential designs 
for maximizing the sum of n observations, Annals of Mathematical 
Statistics, gi, 1060-1070. 
Colton, T.(1963). A model for selecting one of two medical treatments, 
Journal of the American Statistical Association,~, 388-400. 
Cornfield, J., Halperin, M., and Greenhouse, s. w.(1969). An adaptive 
procedure for sequential clinical trials, Journal of the American 
Statistical Association, 64, 759-770. 
Fabius, J. and van ZWet, W. R.(1970). 
Annals of Mathematical Statistics, 
Some remarks on the two-armed bandit, 
41, 1906-1916. 
Feldman, D. (1962). Contributions to the "two-armed bandit" problem, 
Annals of Mathematical Statistics, 13., 847-856. 
Kelley, T. A.(1974). A note on the Bernoulli two-armed bandit problem, 
Annals of Statistics, 2, 1056-1062. 
:::I 
Robbins, H. E.(1952). Some aspects of the sequential design of experiments, 
Bulletin of the American Mathematical Society,~, 527-535. 
't 
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Simon, R., Weiss, G. H., and Hoel, D. G.(1975). Sequential analysis of 
binomial clinical trials, Biometrika, 62, 195-200. 
Smith, c. V. and Pyke, R.(1965). The Robbins-Isbell two-armed bandit problem 
with finite memory, Annals of Mathematical Statistics, .3.2,, 1375-1386. 
Zelen, M.(1969)~ Play-the-winner rule and controlled clinical trials, 
Journal of the American Statistical Association, 64, 131-140. 
(4, 1) 
(6, 2) 
(1, t) 
(2. 1) 
(¼, t) 
(1. 1) 
(2, 2) 
(4, 4) 
(6, 6) 
(1, 2) 
(i, 1) 
(2, 6) 
(1, 4) 
-17-
TABLE 1. Values of r(JL), a(iL), ~(ii,) when ~ is the Product of Densities oC the Form (l~.3). 
(4, 1) (G, 2) (1, i) (2, 1) (t, t) (1, 1) (2, 2) (4, 4) (6, 6) (1, 2) (½, 1) (2, 6) (1, 4) 
.62 
.88 
.70 
,!>O 
.83 
.6·1 
.60 
.84 
.50 
.53 
.79 
.45 
.50 
.83 
.51 
.67 
.86 
.:;A 
-~ 
.ao 
.53 
.53 
.86 
-n 
.50 
.ao 
.53 
.77 
:i 
·.77 
·g4 
• 3 
.73 
.30 
.80 
.83 
.42 
.b 
.80 
.40 
.67 
.75 
.38 
.50 
.75 
.38 
.50 
.67 
·33 
.BG 
.83 
.1~6 
.82 
.78 
.4li 
.68 
.81 
.45 
.70 
.76 
.43 
.50 
.78 
.43 
.50 
.70 
.40 
.50 
.63 
.37 
.89 
.83 
.48 
.87 
.78 
.4"/ 
.69 
.82 
.47 
.72 
.77 
.46 
.50 
.Bo 
.47 
.50 
.72 
.44 
.50 
.65 
.42 
.50 
.6o 
.41 
.89 
.78 
.48 
.70 
.83 
.48 
.73 
.77 
.47 
.50 
.so 
.48 
.50 
.73 
.46 
.50 
.0.5 
.45 
.50 
.61 
,43 
.93 
.82 
.31 
.76 
.29 
.8j 
,72 
.28 
,62 
.70 
.27 
.67 
.63 
.25 
.70 
.57 
.24 
.72 
.54 
.23 
.73 
.53 
.23 
.50 
.47 
.20 
.89 
.82 
:~ 
.76 
.25 
.79 
.75 
,25 
.80 
.71 
1,24 
.63 
.67 
.22 
.67 
.60 
.20 
.69 
.56 
.19 
.70 
.53 
.18 
.70 
.52 
.17 
.53 
.43 
.14 
.50 
.50 
.17 
.99 
.81 
.25 
.99 
.75 
.2t 
,6'( 
.69 
.23 
.75 
.61 
.22 
.87 
.53 
.22 
.89 
.52 
.22 
.52 
.55 
.21 
.50 
.33 
.1 
.99 
.81 
.20 
.99 
.75 
.19 
.89 
.73 
.18 
.93 
.69 
.18 
.72 
.65 
:M 
.54 
.17 
.89 
.52 
.17 
.91 
.51 
.17 
b1 
.43 
.14 
.59 
.50 
:gg 
.30 
.12 
.50 
.29 
.11 
-18-
', 
TABLE 2. Expected Proportion of Success when a1 = b1 = a2 = b2 = 1. 
Optimal 
N Procedure B{~} 
1 -20000 •20000 
2 -2416:Z: -2416:z: 
3 -22226 •22226 4 .569.!!:4 .56244 
2 .57778 .57611 6 .58412 .58403 
1 .52028 .58812 
8 
-22424 •22346 
2 .52866 .52625 
10 .60218 .6001:z: 
12 .61410 .61046 
20 .62126 .617:46 
22 .62612 .62162 
30 .63066 .62212 
32 .63g71 .62'I43 40 
-~ 17 .63410 60 • 271 .63470 
80 
.6462i .63~7 
100 .6421 .639 3 
. 
-.. 
-19-
'• 
TABLE 3. Vnlues of IIUI.X(Ep1 ,F.pp), the Exp~ctcd Proportion o( Success Using B(1,1,) and an Optimal Procedure oc,::.= 
for N ., 25, nnd E max(p11 P:!) for Distributions IJ, as in Table l. 
(a2,b2) (1~ 1 l) (6. :.?) (1. ~) (2, l) (i. t) (1 1 1) (2, 2) (4. 1.) (6. 6) (1, 2) (i, 1) (2. 6) (1, 4) 
(al,bl) 
.86ooo .roooo • eoooo .86000 ,c,JOOO .Boooo .80000 .86000 .80000 .80000 .80000 .00000 .80000 . 
.85800 .8351•9 .83(M .82651 -~1489 .8o746 .80320 .8oo·io .79993 ,r{968l .79953 .79702 .79728 (4,1) .85941 .836.,9 .811-435 .8?.895 .82216 .81o65 .805:~o .80311 .80259 .80087 .80291 .80003 .Boool 
-~9 .SC·jG. • §I1l:',8 .a~111 • -~•m .sun -~111-3 .8111-14 .8112!; .8o9~2 .81818 .8ol22 .801~2 
.75000 .75000 .·,5000 .75000 .'(5000 .75000 .75000 .7~000 .7~000 .75000 .75000 -7~000 
•7'1(15 .79559 .·rB3o8 .70741 .·r5·76o .75227 .74986 .74905 .74729 .7l1-9()6 .74821 -71•792 (6, 2) .79796 .81520 .78939 .73229 .76275 .75461 .752o8 .75153 .75044 .75259 .75000 .75000 
.83orr 
-~61 .822;:i2 .s1212 
-'f?J/1 ·U,fl;6 -~l -~?. -~Glll -~2~~ ·!21~8 -7~~2 • b7 .G66G·r .1:6661 . . (:,'{ .ou 7 • ' 7 • 7 • 0667 • co 7 • 6'Jb7 .G ·r 
.7~84 .78199 .74350 · .7320'l .72133 • 71339 .71045 .68892 .69591 .67067 .668?.l (1, t) .80052 .·,8742 .75190 .73651 .72€1)7 • 7 .. ~105 .71966 .69245 .70091 .67655 .67320 
\al 
.s33:n .Bl~ .7829'1-
-~'.7 :€tit9 .71~2 .74260, ·ij429 -~3 .68922. .6'34R5 .66 7 .66667 • o'f .(5( 7 .66667 • 66'7 • 7 .66£6·, .66067 
.7Go29 .72345 .71111 .70009 .69323 .69037 .67510 .68192 .6(J.sl .66377 (2, 1) .76351 • 7369'• .71595 .70239 .69603 .69415 .67831 .68706 .66890 .66799 
.80000 .rzos3 -12000 -ffm .72222 ·11122 .70000 .71422 .6:zrra .67619 
... .50000 .50000 .50000 .50000 .50000 .50000 .50000 .50000 .50000 
.66001 .63763 .61956 .60656 .©171 .56794, .58028 .52956 .52334 Ci, t> .66772 .65290 .61.270 .63726 .63555 .571.19 .58530 .54006 .52837 
•lfJ.2.E,'f .68120 .6:z21B .668'r.3 ,66232 .6o41:z .61628 -~6212 -~4222 
.50000 .50000 .50000 .50000 .50000 .50000 .50000 .50000 
.62162 .6oo65 .58563 .57961 .54643 .55616 .51549 .51024 (1, l) .62679 .61148 .60310 .6oo63 .54957 .56461• .51977 .51311 
.6666:z .6~000 .61882 .6:f!62 
-~81Jl .€1)000 -2416:z -:m:n 
.50000 .50000 .50000 .50000 .50000 .50000 .50000 
.58407 .56732 .56021 .52752 .5368S .50418 .50136 (2, 2) .58667 .57430 .57035 • 53107 .55013 .50672 .50408 
.~82:z .6ll0J .60:z62 
-~666I -~~jO • 2~!~r£6 ,2n43 
.50000 .50000 .50000 .50000 .50000 .50000 
.55338 .54578 • 51417 .52321 .1.9859 .49700 (4, 4) 
-551•43 .51.859 .51980 .54427 .50121 .50063 
-~2218 -2~22 -~190~ ·2122~ -~1~13 -~1414 
.50000 .50000 .50000 .50000 .50000 
.53953 .50817 .51801 .49750 .49703 (6, 6) .54010 .51650 .54257 .50025 .50011 
-~1823 -~2128 -m2l -21212 -21124 
.33333 .33333 .3~333 .33333 
.42292 .43630 .36880 .35864 (1, 2) .42613 .44893 .37297 .35993 
.4li66:z .482:zl .40~~6 .J9048 
.33333 .33333 .33333 
(l, 1) .45734 .38·ra1 .37~7 .46281 .39673 .37 5 
-~0000 .426~ .4QI22 
.25000 .25000 
(2, 6) .29328 .27438 
.29397 .27869 
. 0 • 1 64 
(1, 4) 
-20-
TABLE 4. Vulucs oC m:uc(E1,1 ,F.p~l, 'th~ l~x1,t1ctcd Proportion oC Success Using B(µ) and an Optimal Procedure 
for N = 50, nnd E nnx(pl'p:~} for Dbtributions l,L as in Table 1. 
m = 
(a?.,b2 ) (1•, 1) (6, 2) (1, ½) (2, 1) (½, ½) (1, 1) (2, 2) ('•· 4) (6, 6) (1. 2) {½. 1) (2. 6) (1, 4) 
al,bl 
.80000 .80560 .80000 .&)00() .80000 .80000 .80000 .Boooo .Boooo .Boooo .Boooo .86000 
.86548 .81•174 .84371 .83-"95 .81191 8ot'.ior: .80?.26 .8o115 .79871 .80233 .79694 .79718 . ::, (4, 1) .86810 .84376 .851128 .83688 .81653 .8o862 .8o516 .8o422 .So215 .8o614 .8oo12 .80009 
.88889 .86164 .8'(380 .8,714 .a·m·1 .8:~143 .81414 .81151. .8c>952 .81818 .8ol52 .E'°l'.i9 
.·,5000 .75000 .·r:,ooo .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 
.8ol1-96 .804511- -791?.2 .76317 .75'505 .75191 .75046 0 71•903 .75248 .74792 .74&J7 (6, 2) .80661 .&?398 -19'-)0'.", .77o62 .75880 .75418 .75306 .75164 .75681 .75003 .75002 
.83orz -~61 :.6,:':•:"!2 16 • • r; 6 . 6 • 6212 • 6111 . 2r.5 128 . 1 2 
. -m-..... , .f:IJ'".)6'( ."J'( 1 7 .66067 ., ., 
.8o375 .79035 .7392·, .72·,93 .71574 .69353 .70163 .67343 .67071 (1, ½) .81195 •7'Rt:Jl- .74607 .734-1•8 .72632 .69872 .70867 .67982 .67w9 
.83333 
-~O • 6(.,(j• .7r:, • 4260 • 142 • 26o .68 _2 .68118 
• 1 . ,oo'( • 7 (i7 7 7 7 007 :i.J'( 
.76987 .73197 .71910 .70726 .69557 .67977 .68758 .66611 .66512 (2, 1) .77500 .74867 .72636 .71109 .70035 .68375 .69498 .67059 .669112 
.80000 .noB·1 .75ooo 
-733U -72::?'c!2 .71795 .70000 .711.29 .67778 .67619 
.50000 .50000 .500t>0 .50.:.>00 .50000 .50000 .50000 .50000 .50000 
.67014 .64-857 .631511- .61926 .614(:e .57547 .588ol .53525 .52786 c,. u -~ .(&56 .65335 .64697 .64520 .58372 .59551 .54665 .53454 
,70'JJ:II 
.68750 .61~:ca .66823 .66539 .€io417 .61628 .56219 -5'•9:':?2 
.50000 .50000 .50000 .50000 .50000 .50000 .50000 .50000 
.63257 .61108 .59752 .59161 .55460 .56539 .52034 .5141.6 (1, ~) .63993 .62334 .61363 .61070 .55974 .57(i01 .5256o .51836 
.66667 .65000 .6·~889 .634&~ .58333 .60000 .54167 .53333 
.50000 .50000 .50000 .50000 .50000 .50000 .50000 
0 59'•93 0 5TI99 .57082 .53541• .54615 .50785 .50453 (2, 2) .59909 .58509 .58o44 .5411::? .56209 .51080 .507CO 
.62857 .613~ .(i0769 .5€,667 .5ano .52576 .52143 
.50000 .50000 .50000 .50000 .50000 .50000 
.56306 .55505 .52154 .53175 .50071 .49933 (4, 4) .561~96 .55811 .52943 .55478 .50318 .5022G 
.59518 .58'752 -5'(905 .57905 -515n .51414 
.50000 .50000 .50000 .50000 .50000 
.44810 .51475 ~52625 .49876 .49838 (6, 6) .54918 .52599 .55235 -~0121 .50088 
.5'7893 .55128 .57593 .51212 .51154 
.33333 .33333 .33333 .33333 
(1, 2) .43~3 .44720 .37695 .36573 
.43 5 .46064 .38194 .36832 
.46667 .1.8571 .1i05'56 .39048 
.33333 .33333 .33333 
(½, 1) .76726 .39561 .38082 
.47540 .40573 .38702 
.50000 .42694 .40722 
.25000 .25000 
(2, 6) .30174 .28269 
.30341 .28803 
. 0 • 1 61• 
.20000 
(1, 4) .26154 
.26377 
.2888