Greenwood press

heart attack
no heart attack
TOTAL
aspirin
147
10,890
11,037
placebo
146
10,888
11,034
TOTAL
293
21,778
22,071
Expected frequencies for the aspirin–physician heart study The computations
assume that the totals represent the population, and that heart condition is inde-
pendent of medication.
The statistician conducts a chi-square test to compare the actual frequencies
to the expected frequencies. In this case, the chi-square indicated that the ob-
78
PROBABILITY

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served frequencies were not close to the expected values, so aspirin reduced heart
attacks.
Making careful lists and working from simple examples can determine many
probability problems. How many families with three children have exactly two
boys? If boys and girls are equally likely, you can list eight possibilities: BBB,
BBG, BGB, BGG, GBB, GBG, GGB, and GGG. The list is called the sample
space, because each family is equally likely. Three of these, BBG, BGB, and
GBB, represent two boys and one girl. So the probability of a family of three chil-
dren having exactly two boys is three-eights, or 37.5 percent. 
The problem of finding how many families would have two boys in three
children can be approached through a simulation. A simulation replaces the ele-
ments of this problem with repeated trials of an experiment using objects that
behave like the birth of children. Tossing a coin could represent the birth of a
child. If you were to determine boys by the head of the coin showing, you could
simulate a family of three children by tossing three coins, say a penny for the first
child, a dime for the second child, and a quarter for the third. This experiment
can be carried 500 or more times very quickly. The probability of two boys
would be estimated by the proportion of times the three coins showed exactly
two heads. In one experiment, this proportion turned out to be 35.8 percent,
which is a little less than the value computed from the sample space. It is now
common to use computers to model complex relationships with simulations.
Computers can generate random numbers (or numbers that act randomly) and
perform rapid computation of probabilities. The Defense Department uses simu-
lations to evaluate outcomes of military actions. Aircraft designers use computer
simulations of air molecules hitting the surface of an airplane to determine its
most efficient shape. The Centers for Disease Control uses simulations to predict
the paths of epidemics. It makes recommendations for vaccinations and preven-
tion procedures based on the outcomes of its simulations. 
Coins and children present examples of binomial probability situations.
When there are two outcomes of a single trial (heads or tails on one coin, boy or
girl for one birth), and a fixed number of independent trials, the computation of
outcome probabilities can be generated by terms in the expansion of the binomial
(p + q)
n
, where 
n is the number of trials, p is the probability of one outcome
(called the success), and 
q = 1 − p is the probability of failure. Families of three
children would be modeled by 
(p + q)
3
= p
3
+ 3p
2
q + 3pq
2
+ q
3
.
The term 
3p
2
q would represent the probability of two boys and one girl. Since 
p = q =
1
2
, the value 
3p
2
q =
3
8
agrees with our previous computation.
The binomial probability theorem provides direct solutions for problems that
don’t have equal probabilities such as the proportion of recessive genes in a pop-
ulation or how many people should be booked for flights so that there are no
empty seats. In a situation in which there are different percentages of a dominant
gene A and a recessive gene a, shouldn’t the dominant gene eventually “win out”