Acceptance Percentage Totals for the University
Chart 2 Accepted Rejected Total Percent Accepted Women 450 550 1000 45% Men 175
325 500 35% Total 625 875 1500 This table shows women actually having a higher
overall acceptance rate than men. This is an example of Simpson’s Paradox
because it involves misleading data. Obviously, the presentation of the data is
very important, and can lead to incorrect assumptions if the data are not used
properly (Internet 2). Paradox 2: An Arrow in Flight One can imagine an arrow in
flight, toward a target. For the arrow to reach the target, the arrow must first
travel half of the overall distance from the starting point to the target. Next,
the arrow must travel half of the remaining distance. For example, if the
starting distance was 10m, the arrow first travels 5m, then 2.5m. If one extends
this concept further, one can imagine the resulting distances getting smaller
and smaller. Will the arrow ever reach the target? (Internet 3) The answer is,
of course, yes the arrow will reach the target. Our common sense tells us so.
But, mathematically, this fact can be proven because the sum of an infinite
series can be a finite number. The question contains a premise, which implies
that the infinite series will result in an infinite number. Thus, 1/2 + 1/4 +
1/8 + ... = 1 and the arrow hits the target (Internet 3). Paradox 3: Two Equals
One? Assume that a = b. (1) Multiplying both sides by a, a² = ab. (2)
Subtracting b² from both sides, a² - b² = ab - b² . (3) Factoring both sides, (a
+ b)(a - b) = b(a - b). (4) Dividing both sides by (a - b), a + b = b. (5) If
now we let a = 1 = b, we conclude, from step (5), that 2 = 1.
Or we can subtract
b from both sides and conclude that a, which can be taken as any number, must be
equal to zero. Or we can substitute b for a and conclude that any number is
double itself. Our result can thus be interpreted in a number of ways, all
equally ridiculous. The paradox arises from a disguised breach of the
arithmetical prohibition on division by zero, occurring at Step (5). Namely,
since a = b, dividing both sides by (a - b) is dividing by zero, which renders
the equation meaningless. As Northrop goes on to show, the same trick can be
used to prove, for example, that any two unequal numbers are equal, or that all
positive whole numbers are equal (Internet 4). Paradox 4: Squares and Rectangles
The area of the square, shown above, is 8 x 8 = 64 units². The square is cut in
the four parts A, B, C, and D, which are rearranged into the rectangle shown
below. This rectangle, however, has an area of 13 x 5 = 65 units². This can lead
to the potential of making 65 units² of gold out of only 64 units². How can you
justify this transformation in area and the creation of matter? The picture of
the rectangle is deceptive! The line XY shown in the picture of the rectangle
(see above) is not a line at all. The parts XU and VY have a gradient of 2 / 5 =
0.4, and the parts XV and UY have a gradient of 3 / 8 = 0.375. So, in fact, XUYV
is a parallelogram with an area of 1, not a line! Paradox 5: Where Is The
Missing Dollar? Three people check into a hotel. They pay $30 to the manager and
go to their room. The manager remembers that the room rate is $25 and gives $5
to the bellboy to return. On the way to the room, the bellboy reasons that $5
would be difficult to share among three people so he pockets $2 and gives $1 to
each person. Now each person paid $10 and got back $1. So they each paid $9,
totaling $27.
