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I have the following template for a journal article featuring a basic table:

The code looks something like this:

documentclass[jou,apacite]{apa6}

usepackage[utf8]{inputenc}

usepackage{amsmath}       



title{Title of the article}

author{Author Name}

affiliation{Affiliation}



abstract{This is the abstract.}



begin{document}



    maketitle    

    section{This is the heading for the first section of the article.}

        Lorem ipsum.



        Results are presented in Table~ref{tab1}.



        begin{table}[!htb]

            caption{Sample table.}label{tab1}

            begin{tabular}{ccc}

                hline\[-1.5ex]

                AAA & BBB & CCC \[0.5ex]

                hline\[-1.5ex]

                1.0 & 2.0 & 3.0\[0.5ex]

                1.0 & 2.0 & 3.0\[0.5ex]

                hline

            end{tabular}

        end{table}



    section{This is the heading for the second section of the article.}

        Lorem ipsum.



end{document}

What I want to know is:
Rather than typing in every value of a table manually, and potentially mistyping a digit or a decimal point here or there; is LaTeX capable of evaluating the math for me?

Consider the odds or probability of a coin toss winning streak, for example:

Each flip has 50:50 (1:1) odds:
50% probability of winning
50% probability of losing

The odds of winning two consecutive flips are 75:25 (3:1):

25% probability of winning
75% probability of losing

The odds of
winning three consecutive flips are 87.5:12.5 (7:1):

12.5% probability of winning

87.5% probability of losing

The odds of winning four consecutive flips are 93.75/6.25 (15:1):

6.25% probability of winning

93.75% probability of losing

..and so on.

Probability of winning n consecutive flips is 2^(-n) * 100.

Probability of not winning n consecutive flips is 100 - (2^(-n))
* 100

Of course I could just calculate each value and type in every cell manually, but I might want to show ten or more flips. Anyway, is this something LaTeX can do?

Here's a LuaLaTeX-based solution to the problem of printing a table of run probabilities -- if the underlying experiment is tossing a "fair" coin.

% !TEX TS-program = lualatex

documentclass[jou,apacite]{apa6}

usepackage{amsmath,booktabs,lipsum}  

letPrrelax % undefine "Pr"

DeclareMathOperator{Pr}{Pr}



usepackage{unicode-math} % choose suitable math and text fonts

setmainfont{Stix Two Text}[Ligatures={TeX,Common}]

setmathfont{Stix Two Math}



title{Title of the article}

shorttitle{Title}

author{Author Name}

affiliation{Affiliation}

abstract{This is the abstract.}



usepackage{luacode}

begin{luacode}

function run_prob ( j )

   return (0.5)^j

end

function printrows ( n )

   for i=1,n do

      runprob     = 100*run_prob(i)

      runprobcomp = 100-runprob  

      tex.sprint ( i .. "&" .. runprob .. "&" .. runprobcomp .. "\\" )

   end

end

end{luacode}



begin{document}

maketitle    



section{In the beginning}

Lorem ipsum.



Results are presented in Table~ref{tab1}.



begin{table}[!htb]

caption{Fair coin tosses: Probabilities of runs of length $n$, in percent.}label{tab1}



$begin{array}{@{}lll@{}}

toprule

n & Pr(text{Run}) & 100-Pr(text{Run})\

midrule

%% Invoke the 'printrows' Lua function:

directlua{printrows(12)}

bottomrule

end{array}$

end{table}



lipsum[1-3]

end{document}

A solution using the CAS, Sage, given below. The documentation for sagetex on CTAN is here and an example using Sage/LaTeX/sagetex is here. Sage is not part of the LaTeX distribution so it needs to be installed on your machine or open a free Cocalc account (the easiest way). Here is the code.

documentclass[jou,apacite]{apa6}

usepackage[utf8]{inputenc}

usepackage{amsmath}       

usepackage{sagetex}

title{Title of the article}

author{Author Name}

affiliation{Affiliation}

abstract{This is the abstract.}



begin{document}

begin{sagesilent}

output = r"begin{table}[!htb]"

output += r"caption{Sample table.}label{tab1}"

output += r"begin{tabular}{ccc}"

output += r"hline"

for i in range(1,13):

    output += r"%s & %s & %s \"%(i,(100*.5^i).n(digits=7),(100-100*.5^i).n(digits=7))

output += r"hline"

output += r"end{tabular}"

output += r"end{table}"

end{sagesilent}



maketitle

section{This is the heading for the first section of the article.}

The probability of $6$ consecutive flips is given by $sage{(100*(.5)^6).n(digits=7)}$%.

The probability of $7$ consecutive flips is given by $sage{(100*(.5)^7).n(digits=7)}$%.



Applying this to other values we can create the Table below:

Results are presented in Table~ref{tab1}.

sagestr{output}

section{This is the heading for the second section of the article.}

Lorem ipsum.

end{document}

The output running in Cocalc is shown below:

In order to create the table with a loop, Sage has to run. When you build your file in Cocalc, LaTeX runs, then Sage, then LaTeX (where the results from the Sage run are inserted). This means in order for the first pass of LaTeX to compile, the table is not there. It gets created as a string in Sage and then inserted for the final LaTeX compilation. sagestr{} gets string data as opposed to sage{} for numerical data. Two nice features that you get with Sage: the ability to do calculations on the fly such as with the line The probability of $6$ consecutive flips is given by $sage{(100*(.5)^6).n(digits=7)}$%. where digits=7 sets the number of significant figures to use in the answer and Sage also can handle numerous probability distributions, see here. Finally, Sage allows you to use Python in your coding. With a CAS, Python, and LaTeX you can do pretty much anything.