In 2011, Donald Smith from Victoria University asked me a question about computing gambling outcomes using Mathematica. I generated a CDF examining three cases.
Probability of correctly selecting 1 from m from n goes, with vertical lines at the expected average, and the break-even line when paying odds of p:1:
For larger samples the abscissa indicates percentages, which keeps the average and break-even lines in the same place for comparison purposes:
Use RandomVariate to randomly select 10k samples from BinomialDistribution, plotting the results and expected results onto percentage abscissa: