The purpose of this part of the website is to provide greater depth of the certain mathematical developments of some chapters of the book. Please contact the author at faithequation@gmail.com if you have further inquiries.
Chapter 4: The Issue of Independent Events. An opposition might arise in the mind of the reader with a background in the theory of probability and statistics. It concerns the assumption of independent events when obtaining the overall probability of the 9 prophecies being fulfilled by multiplying the 9 individual prophecies. One might reasonably assume that if one prophecy has been fulfilled, it might increase the likelihood that a second prophecy will be fulfilled, and if two prophecies have been fulfilled, it is even more likely that a third prophecy will be fulfilled. In general, does the fact that some prophecies are known to have been fulfilled increase the probability that subsequent prophecies will be fulfilled? We consider this possibility using an argument based on conditional probability.
1) Given that one prophecy has been fulfilled, the conditional probability that another prophecy will be fulfilled increases by an arbitrary percentage q1(100%), 0 ≤ q1 ≤ 1, relative to the unconditional probability of fulfilling this prophecy. Thus, if one prophecy has occurred, and the unconditional probability of the second prophecy is, the conditional probability that the 2nd prophecy will be fulfilled given that the first prophecy has been fulfilled is p2(1 + q1).
Note: Since q1 = the increase in conditional probability of the 2nd prophecy occurring (given that the 1st prophecy has occurred) relative to the unconditional probability of fulfilling this 2nd prophecy (i.e., p2), the actual increase in the conditional probability (over the unconditional probability) is p2q1. Thus the conditional probability that the 2nd prophecy will occur given that the first has occurred is
p2 + p2q1 = p2(1 + q1).
Note that
Thus, the joint probability that the first two prophecies are fulfilled is
p1p2(1 + q1)
2. Given that two prophecies have been fulfilled, the conditional probability that a third prophecy will be fulfilled increases by q2(100%), 0 ≤ q2 ≤ 1, relative to the conditional probability of this prophecy being fulfilled given that only one prophecy has been fulfilled. Thus, if denotes the unconditional probability that the 3rd prophecy is fulfilled, and p3(1 + q1) denotes the conditional probability that the 3rd prophecy is fulfilled given that the 1st prophecy has been fulfilled, then p3(1 + q1)(1 + q2) is the conditional probability that the 3rd prophecy will be fulfilled given that both the 1st and 2nd prophecies have been fulfilled. Note that
Thus, the joint probability that the first three prophecies are fulfilled is
p1p2(1 + q1)p3(1 + q1)(1 + q2) = p1p2 p3 (1 + q1)2(1 + q2).
3. Now given that three prophecies have been fulfilled, the conditional probability that a 4th prophecy will be fulfilled increases by q3(100%), 0 ≤ q3 ≤ 1, relative to the conditional probability of this prophecy being fulfilled given that only two prophecies have been fulfilled. Thus, if p4 denotes the unconditional probability that the 4th prophecy is fulfilled, and p4(1 + q1) denotes the conditional probability that the 4th prophecy is fulfilled given that the 1st prophecy has been fulfilled, and the conditional probability that the 4th prophecy is fulfilled given that the 1st and 2nd prophecies have been fulfilled is , p3(1 + q1)(1 + q2) then p4(1 + q1)(1 + q2)(1 + q3) is the conditional probability that the 4th prophecy will be fulfilled given that the 1st, 2nd and prophecies have been fulfilled. Note that
Therefore, the joint probability that the first 4 prophecies are fulfilled is
p1p2(1 + q1)p3(1 + q1)(1 + q2)p4(1 + q1)(1 + q2)(1 + q3)
= p1p2p3p4 (1 + q1)3(1 + q2)2(1 + q3).
Extending this argument we see that the conditional probability of the nth prophecy being fulfilled given that the previous n−1 prophecies have been fulfilled is
pn(1 + q1) (1 + q2)………. (1 + qn-1),
where denotes the unconditional probability of the nth prophecy being fulfilled.
Thus, by the Multiplication Rule, the probability of n specific prophecies being fulfilled is
p1 p2(1 + q1) p3(1 + q1)2 (1 + q2) …pn(1 + q1) (1 + q2)…(1 + qn-1)
= p1 p2 …… pn (1 + q1)n-1(1 + q2)n-2 ……… (1 + qn-1).
Remark: Technically, since probabilities cannot exceed one, we should write the above conditional probability as
Min{1, pn(1 + q1) (1 + q2)………. (1 + qn-1)}.
However, since typically is quite small, we can reasonably assume that pn
pn(1 + q1) (1 + q2)………. (1 + qn-1) < 1.
Now let
Q = Max{q1, q2, ………., qn-1}.
Remark: Presumably as more and more probabilities are fulfilled, the conditional probability of the next one being fulfilled increases. Thus, it would appear that Q = qn-1, but from a purely mathematical point of view, this need not necessarily be the case.
Now
p1 p2 …… pn (1 + q1)n-1(1 + q2)n-2 ……… (1 + qn-1)
≤ p1 p2 …… pn (1 + Q)(1 + Q)2 ………. (1 + Q)n-1
= p1 p2 …… pn (1 + Q)1 + 2 + …… + n-1 = p1 p2 …… pn (1 + Q)n(n-1)/2.
In the case of 9 prophecies,, and the probability that all 9 are fulfilled is
p1.p2.p3.p4.p5.p6.p7.p8.p9 (1 + Q)36 .
Now, if we were to assume that Q = 0.10, which in this context is relatively high, the conditional argument increases the probability that 9 prophecies are fulfilled by
(1.1)36=30.913<100=102
But, as already computed in Chapter 4,p1⋅p2⋅p3⋅p4⋅p5⋅p6⋅p7⋅p8⋅p9=10−76 , and 100X10-76=10-74 so the probability of these prophecies being fulfilled is still less than 10(i.e., negligible). We reach the conclusion that any increase in the overall probability of these 9 prophecies being fulfilled is negligible, and that the assertion of independent events allowing multiplication of probabilities is valid.
Remark: Even if we assume Q = 0.2,
(1.2)36=708.8<1000=103.
But since
p1⋅p2⋅p3⋅p4⋅p5⋅p6⋅p7⋅p8⋅p9=10−76,
p1⋅p2⋅p3⋅p4⋅p5⋅p6⋅p7⋅p8⋅p9 (1 + Q)36 10–76 × 103 = 10–73,
which is still negligible.
If the reader chooses to reject this reasoning, he/she should cling to the idea that the Bible contains hundreds if not thousands of prophecies which have come true, with none failing. That is staggering evidence that we should pay expect the other prophecies to come true, and that the Bible is a reliable document.
Chapter 5: The Mathematics of the Curve Fitting – Advanced Approach
For more mathematical depth about how the models in Chapter 5 were formed, let’s retrace the curve fitting from a more advanced statistical approach, using calculus and statistical theory. Look again at the data in Table 5.1. Suppose we try to fit the data to a polynomial predicting function. To decide the appropriateness of this decision, consider the error function defined by
ErrorFcn = Actual Data – Predicted Values
We do a scatterplot of the ErrorFcn versus time, as follows.
Note that the function values tend to spread out as time increases. This indicates that by taking the natural logarithm of the data, we have can use a polynomial as a predicted function. When we take the natural log, the errors have to be constant, or within a bound. Taking the ln is a reasonable choice for an error stabilizing transformation, since growth tends to be exponential.
We take the ln of the data and form a scatterplot as follows.
We notice a “bump” in the data around 1350 AD where the population decreases. To better determine this “bump” we do a transformation of the data, considering the annualized growth rate:
We do this annualizing because the distance between data points is not equal. To get the distances between data more equal we make this transformation. Then the bump point is vivid between 1200 and 1350. That “bump” point occurs around the time of the Bubonic Plague.
BUBONIC PLAGUE: The Black Death, or Bubonic Plague swept Europe and parts of Asia in the 14th century, killing as much as 75% of the population of Europe and parts of Asia in less than 20 years.
Because of these analyses we decide to restrict our data to 1350 AD and beyond.
To discover the form of the curve, note the following table regarding polynomial functions for curve fitting.
| Degree of Polynomial | Number of Data Points Needed |
| 2 | 3 |
| 3 | 4 |
| n | n+1 |
Thinking of growth rate as the derivative of a polynomial, if we have n+1 data points, then a polynomial of degree n gives a perfect fit. Suppose we have 100 data points, then we know a polynomial of degree 99 will fit growth perfectly. But a statistician-mathematician would look for a more “elegant” fit by finding a polynomial of smaller degree if possible, especially in order to compute function values. To make this analysis, we consider the increasing pattern of r2, where
From statistical theory, it is know that
lim r2=1
(As the degree in the regression poly ⇒ n+1)
Here, the cubic fit has r2=0.973, but quadratic fit has r2=0.943. Since there is only a small percent change in the difference 1−r2 from a cubic to a quadratic, we consider a cubic function for growth rate. Then the actual fitted function would be its antiderivative, a quartic polynomial. Therefore, we choose the fit for world population to be
ln(World Population) = Quartic Function of Time, in Years
Hence, the fitted world population function is given by
Fitted World Population = e(Quartic Function of Time, in years)
Such thinking yields similar models for the Evangelized and Christian Populations. From here, we proceed in the modeling as in the book.