Random Digit Table vs Simulation

Two properties define the table: each position is equally likely to hold any of the ten digits, and the positions are independent of one another. Everything about using one follows from that. Three things have to be fixed before you read anything: labels that all carry the same number of digits, a stated starting point, and a stated skip rule. Equal label length is what keeps every individual equally likely.

Three students are wanted from a class of 40. Label them 01 to 40, read two digits at a time, skip any group above 40 or equal to 00, and skip any label already taken. Starting at a line reading 9 2 3 5 2 3 3 5 1 1, the groups are 92, discarded as out of range, then 35 and 23, both taken, then 35 again, discarded as a repeat, then 11, taken. The sample is students 11, 23 and 35. Expect heavy discarding: only 40 of the 100 two-digit groups are usable here, so 60 percent of what you read is thrown away.

"Label them 1 to 40 and read one digit at a time" cannot work, because a single digit only ever produces labels 0 through 9 and students 10 through 40 have no chance of selection at all. Patching it by reading a second digit only when you feel you need one is worse, since the chance of selection then depends on the label in a way that is hard even to write down. Picking the starting line after glancing at the digits, or stopping early because the sample already looks representative, break the mechanism the same way.

The skip-repeats rule is what makes this sampling without replacement. Drop that rule and repeated labels are kept, which is a different design with a different set of formulas behind it.

Topic 1.11 describes carrying out a simple random sample with a random number generator or numbered slips of paper. The table is the printed version of the same chance device, and random digit table vs simulation sets the two mechanisms against each other.

A simulation answers a probability question by generating outcomes from a model you have assumed and then counting. Fix the model, define exactly what one trial is, run many trials, and report $\hat{p}$ (p-hat) as successes divided by trials. What comes back is an estimate of the probability under that model. It is not the probability itself, and it is not evidence the model is true.

Take a player who makes 70 percent of her free throws with shots independent, and ask for the chance she makes all four of her next attempts. The exact answer is $0.70^4 = 0.2401$. One run of 20 simulated trials gave 3 successes, an estimate of 0.15. A separate run of 10,000 trials from the same model gave 2,335 successes, or 0.2335. Same model both times, and the longer run lands closer.

The misreading: my simulation gave 0.15, so the probability is 0.15. An estimate from $n$ trials has standard deviation $\sqrt{p(1-p)/n}$, which at $p = 0.24$ and $n = 20$ is 0.096. Two standard deviations either side of 0.24 spans roughly 0.05 to 0.43, so a 20-trial run landing on 0.15 is unremarkable. Reporting it as the answer treats a number with that much slop as exact.

Precision improves slowly, and the square root is why. The standard deviation falls with $\sqrt{n}$, so halving the error takes four times the trials and cutting it to a tenth takes a hundred times: 0.043 at 100 trials, 0.0043 at 10,000.

The harder limit is the model. That simulation tells you what a 70 percent shooter would do. It cannot tell you whether this player is a 70 percent shooter, and if she is really at 0.60 then every figure it produced answers the wrong question. Topic 2.3 in the Fall 2026 course is Estimating Probabilities Using Simulation.