Probability 5th grade In 1880 population was 1 billion. In 1920 population was 2 billion. In 1980 population was 4 billion.  Following this trend would the population be 6, 8, 10, or 12 billion in 2020? Answer on official answer key is 8 billion. How is this number calculated?
 A: You assume population follows an exponential trend:
$$P = a \, e^{b t}$$
where $P$ is the population (in billions), $t$ is time (in years), and $a$ and $b$ are unknown constants.  You can convert this to a linear equation by taking logs:
$$\log{P} = \log{a} + b t$$
We have an overdetermined system, with $3$ equations and $2$ unknowns
At $t=0$ (1880), $P=1$ at $t=40$, $P=2$; then we have:
$$0 = \log{a} + b (0) $$
$$\log{2} = \log{a} +b (40) $$
$$\log{4} = \log{a} +b (100)$$
Then we perform a least squares by considering the matrix
$$A = \left (\begin{array}&1 & 0 \\ 1 & 40\\1 & 100 \end{array}\right )$$
and vector 
$$b = \left (\begin{array}&0\\ \log{2} \\ \log{4} \end{array}\right )$$
The least squares solution is
$$\left (\begin{array}&\log{a} \\ b \end{array}\right )= (A^T A)^{-1} A^T b$$
so that, sparing you the details, the best fit is
$$P(t) = 2^{(3/38)(1+t/4)}$$
therefore, our best guess is in 2020,
$$P(140) = 2^{(3/38)(36)} \approx 7.17\, \text{billion}$$
so $8$ billion is the best answer. 
A: The population increased by a factor of 4 from 1880 to 1980, and although you don't have any reason to assume it will do the same from 1920 to 2020, you don't really have anything else to go on. So assume it, and conclude the population will be 8 billion.
