Greenwood press

PROBABILITY
81

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Nuclear medicine

Poker probabilities

Probability and utility of the real world

Probability in the real world

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PROPORTIONS
Proportions are equations that compare ratios or scaled quantities. Cartog-
raphers use proportions to make maps, because they need to scaledown distances
so that large pieces of land can be viewed on a sheet of paper. For example, the
state of Illinois is approximately 370 miles long. If the map maker wants to place
Illinois on a sheet of paper that is 25 cm long, a proportion that can be used to
determine a scale in this situation is 
s miles on map
370 miles
=
1 cm on map
25 centimeters
. Cross multi-
plying these quantities helps solve the equation, 
s = 370/25 = 14.8. The legend
on the map might indicate that 1 cm represents 15 miles. 
Eratosthenes used a proportion to determine the radius of the earth in around
230
BC
. He had traveled between the cities of Alexandria and Cyrene and deter-
mined that its distance was around 5,000 stades, where each stade is about 559
feet. At noon, he had measured the angles of shadows formed by sticks in the
ground and determined that there was not any shadow at Cyrene, and an angle of
elevation of 
α = 82.8 degrees at Alexandria, as shown below. 
Eratosthenes argued that the angle formed near the top of the stick, 7.2 degrees,
is the same as the central angle in the earth that determines the sector between
the two cities, since light rays travel parallel to the earth, as shown as follows.