Random uniform numbers on the interval (0,1) are the kind of numbers that are generated by the RAND() function you have on your spreadsheet. The usual RAND() function will be based on an algorithm that uses some form of modular arithmetic and is probably keyed to the clock on your computer. It strives as well as possible to select a completely random number between the two integers 0 and 1. It will be a decimal number, of course, like 0.3752 (you can specify the number of decimal places).
It will be a random uniform number in this sense: RAND() is just as likely to select a number from 0.1000 to 0.1999 as it is to select a number from 0.2000 to 0.2999. That is, the probability it will select a number from the first interval is 1 over 10, the same as the probability (1 over 10) it will select a number from the second, equal-length interval. Looking at this another way, if we restrict the selection to four-decimal numbers ranging from 0.0000 to 0.9999 there will be exactly 10,000 numbers that can be selected. If the selection process is purely random then the probability that any one of those 10,000 numbers will be selected by the RAND() function is exactly 1 over 10,000. This is the "uniform" feature. Each of the four-decimal numbers is as likely to be selected as any other, or we say that the numbers are "equally likely" to be selected.
How could we use the RAND() function to randomly select numbers for a Lotto ticket? Consider for convenience Lotto Texas, where 6 out of 50 integers are chosen to comprise a Lotto ticket. We could tell the RAND() function to generate two-decimal random numbers ranging from 0.00 to 0.99, so that exactly 100 such numbers are equally-likely to be selected. Then if RAND() generates either the number 0.00 or the number 0.01, we could agree that from the 50 integers available for choice on the Lotto ticket the integer selected is 1. If RAND() selects either 0.02 or 0.03, our Lotto integer is 2; if it selects 0.04 or 0.05, the Lotto integer is 3, and so on up to 0.98 or 0.99, which would be associated with the Lotto integer 50. This is quite likely the way it is done by your spreadsheet when you ask it to generate a random integer from 1 to 50 using the RAND(50) function. That is, the RAND(50) function could use the RAND() function to obtain a random uniform number on the interval (0,1), then could convert it to a random integer from 1 to 50 using a conceptual table to match the RAND() number with the appropriate integer. We, of course, could actually construct such a table and use it to do our matching if we had available only the RAND() function and not RAND(50).
That table as you envision it now, where with each of 50 integers is associated two adjacent two-decimal numbers ranging from 0.00 to 0.99, is almost what is meant by the phrase distribution function as it is used on the next page you will study. It would reflect our assumption that each of the balls numbered 1 through 50 would be equally likely to be chosen by the ball-selection process. Thus we would be using what we call a distribution function for the uniform distribution, since the number of times each ball would be chosen over a long series of selections would be about the same, or the chosen numbers would be "uniformly distributed" in regard to the number of times they are chosen.
But not all distribution functions have that uniformity feature. And this is the point were our discussion begins to get interesting as far as its implications for Lotto are concerned. On the next page will begin a lesson on distribution functions in general, where we consider how to randomly select, using RAND(), from distributions that are not uniform. Does that pique your curiosity? How do we use random uniform numbers to select from distributions that are not uniform?
Read on.