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MUTUALLY EXCLUSIVE AND INDEPENDENCE:
UNRAVELLING BASIC MISCONCEPTIONS IN PROBABILITY THEORY
I.W. Kelly University of Saskatchewan
F.W. Zwiers Environment Canada
T w o c e n t r a l concepts in p r o b a b i l i t y t h e o r y a r e those o f "independence" a n d o f "mutually exclusive" e v e n t s a n d t h e i r alternatives. In t h i s a r t i c l e we p r o v i d e f o r t h e i n s t r u c t o r suggestions t h a t can b e used t o e q u i p s t u d e n t s w i t h an i n t u i t i v e , comprehensive u n d e r s t a n d i n g o f these basic concepts. L e t u s examine each o f these concepts i n t u r n along w i t h common s t u d e n t misunderstandings.
1. Mutually Exclusive versus Non Mutually Exclusive Events
Events a r e m u t u a l l y exclusive when t h e occurrence o f one o f t h e e v e n t s r u l e s o u t t h e p o s s i b i l i t y o f t h e occurrence o f t h e other events o f concern. T h e outcomes, f o r example, on t h e toss o f a single die a r e 1, 2, 3, 4, 5 o r
- T h e outcomes a r e all m u t u a l l y e x c l u s i v e because when a d i e i s tossed a n d a number t u r n s up, all t h e o t h e r numbers cannot occur. On t h e o t h e r hand, t h e e v e n t s "eating r o c k y mountain o y s t e r s f o r b r e a k f a s t " a n d "eat- i n g vegetable soup f o r b r e a k f a s t " a r e & mutually exclusive since it i s p o s s i b l e t h a t one has b o t h f o r breakfast, however improbable t h a t m i g h t be.
T h e d i s t i n c t i o n between c o n t r a d i c t i o n a n d c o n t r a r i e t y should b e i n t r o d u c e d a t t h i s p o i n t t o students. T w o statements are contradictories when t h e y cannot b o t h b e t r u e , and cannot b o t h b e false. T w o statements a r e con-
t r a r i e s if t h e y can b o t h b e false a n d a third statement, d i f f e r e n t f r o m
both, can b e t r u e. For example, "It i s r a i n i n g outside t h i s b u i l d i n g a t t h i s moment" a n d "It i s not r a i n i n g o u t s i d e t h i s b u i l d i n g a t t h i s moment" a r e contradictories. However, t h e statements " A l l mathematicians a r e v e r y i n - t e l l i g e n t " a n d "No mathematicians a r e v e r y intelligent" are c o n t r a r i e s since a l t h o u g h b o t h cannot b e t r u e , it i s almost c e r t a i n l y t h e case t h a t b o t h a r e false. Some mathematicians a r e v e r y i n t e l l i g e n t a n d others are not.
T h e r e a r e m u t u a l l y exclusive e v e n t s o f b o t h t y p e s. I n t h e d i e t o s s i n g example a l r e a d y described we c o u l d d i v i d e t h e sample space i n t o t w o parts, say, "even" a n d "odd". These e v e n t s are m u t u a l l y exclusive a n d c o n t r a - d i c t o r y. B y t a k i n g t h e sample space t o b e t h e outcomes 1, 2, 3, 4, 5 a n d 6 we have i m p l i c i t l y made an assumption about t h e operation o f d i e tossing: namely, t h a t it i s impossible f o r t h e d i e t o come t o r e s t on a p o i n t o r an
edge. If t h e sample space were expanded t o include these possibilities t h e n
t h e e v e n t s "even" a n d "odd" a r e contraries. T h e reason f o r i n t r o d u c i n g these concepts is, o f course, t o e n s u r e t h a t t h e s t u d e n t i s q u i t e clear a b o u t t h e f a c t t h a t a p a i r of m u t u a l l y e x c l u s i v e events a r e n o t necessarily complementary events.
The die example presented above has somewhat of a contrived air about it
because most of us would accept the proposition t h a t the probability of a die coming t o rest on a point o r an edge is zero. However, things are not always t h a t clear c u t i n nature. Amongst humans there are males, females, and other borderline cases (e.g., Klinefelter's syndrome) which occur with nonzero probability. Thus, the events "male" and "female" are i n fact con- traries.
Another point that should be considered i n class is the fact t h a t v e r y often
i n the real world it is not always clear whether o r not two events are
mutually exclusive. The examples we present t o students when teaching
t h e concept are, understandably, clearcut. For example, if we obtain a six
on a single toss of a f a i r die, it rules out the possibility of obtaining a
one, two, three, four, o r five. I n the real world we often do not have enough information t o be sure. Not all students are aware t h a t "Clark Kent'' and "Superman" are not mutually exclusive. It would be useful t o draw examples from nature where the distinction between contradictory, contrary, mutually exclusive and non-mutually exclusive are not always clear.
2. Independent versus Dependent Events
Events are independent when the occurrence (or nonoccurrence) of one of the events carries no information about the occurrence (or nonoccurrence) of the other event. Mathematically, two events A and B are considered t o be independent if P(A n B) = P(A)- P(B). For example, if the probability t h a t Obadiah has escargot f o r breakfast tomorrow is 0.4 and the probabil- ity that it will rain tomorrow is 0.3, then the probability that both events will occur tomorrow is (0.4)*(0.3)=0.12. When we have more than two events, the situation becomes a b i t more complicated - all possible combi- nations of component events must follow the multiplication rule. That is, each combination must also involve independent events.
Students have several difficulties with the distinction between independent and dependent events. The f i r s t parallels the problem t h a t arises with mutually exclusive events, namely, determining when events i n the real world are independent o r dependent. Once again, we seldom help students t o bridge the gap between the fuzzy distinctions apparent i n nature and
the v e r y r i g i d distinctions made i n mathematics. If we toss a pair of dice,
the outcome t h a t occurs on one due obviously (for most of us) does not i n - fluence what outcome will occur on the other die. I n other cases one often needs expertise i n a particular area t o make a reasoned judgement whether particular events are independent or not. Many years of research were required t o demonstrate t h a t there is a dependent relationship between smoking and lung disease (see also, Ayton & Wright, 1985).
A second common misunderstanding involves interpreting a dependent rela- tionship between events as a causal relationship. There are, of course, s i t - uations where this is plausible, f o r example, having tuberculosis is depen- dent on having tuberculosis bacilli i n one's body. However, there are many examples of dependent relationships between events where no causal rela- tionship is involved. For example, having a f i r e is dependent on the p r e -
t u n i t y t o revise your bet. You realize t h a t there are only four ways i n which the outcome of six tosses of a coin can result i n a total of three heads with a head on each of the last two tosses. Therefore, the relative chances of being a winner once you've been given some knowledge of the outcome of the experiment has been reduced from one-in-four t o one-in- five. The events "three heads i n six tosses of a coin" and "the last two tosses i n a sequence of six tosses are heads" are not independent.
The element of time i n the above example relates not t o the way i n which the experiment was conducted, b u t t o the way i n which we think. We were given some knowledge about the outcome of the experiment, namely that three heads had occurred. This i n t u r n gives us some knowledge about the likelihood of other events t h a t may have occurred. We now t h i n k t h a t it is less likely that there was a head on each of the last two tosses than before the experiment was conducted. This is essentially what we mean when we say t h a t two events are dependent - knowledge t h a t one event has oc- curred conveys information about whether o r not the second event also occurred. The events "heads on the f i r s t toss" and "heads on the t h i r d toss" are independent, because when we re-evaluate the likelihood that second event occurred i n l i g h t of the occurrence of the f i r s t event, we see t h a t we have no more knowledge about the outcome of the second event.
The element of time which we mentioned is associated with this process of re-evaluation of probabilities after some information about the outcome of the experiment is available. These a posteriori, o r conditional probabil- ities can only be applied t o decision making, such as whether t o continue a bet o r raise the ante i n a poker game, after some information about the out- come of the experiment is available.
These probabilities are evaluated b y conceptually repeating the experiment with a restricted sample space. We feel t h a t students have some inkling of what goes on, b u t t h a t we don't explain these concepts t o them carefully enough. We suspect t h a t they do feel t h a t something happens as time goes on b u t don't really understand its mechanics. We should point out, using simple examples such as coin tossing o r card games, t h a t we construct new probability models for the experiment as it progresses (or after the fact) which are conditional upon information which we have received about the outcome of the experiment t o that point. With these ideas about conditional probability i n place it can be shown t h a t two events, say A and B, are i n -
dependent if the a p r i o r i probability of A is equal t o the a posteri-
o r i probability of A given B. We t h i n k t h a t such an approach is much easier f o r the student t o understand than the standard approach which is found in most t e x t books. The latter consists of stating the usual defini-
tion of independence, i.e., t h a t A and B are independent if
P (A n B) = P (A) P (B)
and then presenting a few examples t o illustrate the t r u t h of the defini- tion. We^ often^ confuse^ the^ student through our^ use of^ language,^ as^ dis- cussed above, and the failure to l i n k the ideas of independence and condi- tional probability.
- Confusion Between Independent Events and Mutually Exclusive Events
Students not only have difficulty with the notions of mutually exclusive and independent events; they v e r y often confuse the two. Most of the con- fusion arises because we, as instructors, do not take the time t o relate the two concepts.
We frequently answer the question "If A and B are mutually exclusive,
does it follow that they are not independent?" With the reply, ,"well, no.
For example,.. ." and then t r o t out the pathological example of a pair of
events of probability zero without f u r t h e r explanation. Except i n certain pathological cases, the concept of mutually exclusive events is the exact
antithesis of the concept of independent events. A nice illustration is p r o -
vided b y Hays (1981, p. 43-44). Suppose that all men are either "bald" o r have a "full head of hair". These are mutually exclusive. Let us say the probability of selecting a bald man from the population is 0.60; the proba-
b i l i t y of selecting a hairy man would be 0.40. If these two events were i n -
dependent, then the probability of selecting a man who i s both bald with a head f u l l of hair would be equal t o (0.60). (0.40). B u t the probability of such a joint event is, of course, zero. Mutually exclusive events are ( a l - most) never independent. Ad hoc explanations tend t o deal with specific aberrations t h a t may lead t o f u r t h e r confusions and moreover avoid dealing with the fundamental principles upon which the student's question i s based.
B y planning f o r instruction of these two fundamental concepts we can i n - sure t h a t the student's understanding i s built up systematically. This p r o - vides a firmer foundation upon which the student can acquire a grasp of probability theory.
References
Ayton P., E Wright, G. (^) (1985). Thinking with probabilities. Teaching sta- tistics, 2(2), 37-40.
Capra, F. (1975). The tao of physics. Berkeley: Shambala Books.
Hays, W. L. (1981). Statistics. New York, N.Y.: Holt, Rinehart & Winston.
Underwood, B. J. (1957). Psychological research. New York, N. Y. : Appleton-Century-Crofts.
