Topic 4.2 Estimating Probabilities Using Simulation: design and carry out a simulation using a chance device or random numbers to estimate a probability as a long-run relative frequency.

What this topic is asking

The College Board (Topic 4.2) wants you to design and carry out a simulation, using a chance device or random numbers, to estimate a probability as a long-run relative frequency, and to interpret what one trial and many trials produce.

What a simulation is

Simulation is the practical companion to Topic 4.1: where 4.1 says probability is a long-run relative frequency, 4.2 says you can therefore estimate any probability by generating that long run artificially. It is especially valuable for complex situations (collecting all prizes, waiting times, multi-step processes) where an exact formula is awkward, but the College Board also uses it to build intuition for the rules that come next.

The four-step method

Every full-credit simulation answer touches all four. The most common omission is a vague stopping rule, so be explicit about when a trial ends (for example, "stop when all four prizes have appeared" or "read exactly five digits, one per patient").

Assigning digits correctly

The heart of a good simulation is the digit assignment, because it encodes the probabilities. If an event has probability $p$ expressed in tenths, assign that many of the ten digits $0$ to $9$ to it; for hundredths, read digits in pairs ($00$ to $99$) and assign the right number of the $100$ pairs. For a probability like $0.3$, any three of the ten digits work (commonly $0,1,2$); the choice of which three does not matter, only that exactly three are used. When a probability does not divide the digits evenly (such as $1/3$), you let some digit combinations be ignored and re-read, keeping the kept outcomes in the correct ratio. Modelling several independent events in one trial just means reading more digits, one block per event, with each block using the same assignment. Getting this mapping faithful to the stated probabilities is what makes the simulation a valid model rather than an arbitrary game.

Reading the estimate

After many trials, the estimate is simply a relative frequency: the number of trials in which the event happened, divided by the number of trials. Because it is a long-run relative frequency, it is an estimate of the true probability and, by the law of large numbers, it improves as you run more trials, which is why exam answers note that "more trials give a more reliable estimate." When the question asks for an expected value (such as the average number of boxes needed), you record a count each trial and average those counts instead of counting successes. Either way, you should interpret the result in context: not "the estimate is $0.42$" but "we estimate about a $42\%$ chance that ..." or "on average about $8$ boxes are needed." This habit of translating the numerical estimate back into the situation is what the College Board rewards, and it mirrors the interpret-in-context discipline that runs through every unit.

Try this

Q1. To simulate an event with probability $0.6$ using single random digits, how many digits should represent the event, and give a valid assignment. [2 points]

• Cue. Six of the ten digits; for example let $0$ to $5$ mean the event and $6$ to $9$ mean not (any six distinct digits work).

Q2. Why does running more trials of a simulation give a better probability estimate? [1 point]

• Cue. The estimate is a long-run relative frequency, and by the law of large numbers it converges to the true probability as the number of trials increases.