BRM Chapter 19 - Simulation

Simulation - definition

Using random digits from a table or computer software to imitate chance behavior according to a specified probability model.​

Why use simulation?

Exact probability calculations are often very difficult, but simulation is easier than math and faster than physically repeating real‑world trials.​

Role of the probability model in simulation

Simulation results are only as good as the underlying probability model; if the model is unrealistic, the simulated probabilities will be misleading.​

Where probabilities come from - three sources

Probabilities can come from long‑run data, personal (subjective) judgment, or from a mathematical model describing the random phenomenon.​

Simulation basics - key idea

Once a trustworthy probability model is specified, many repetitions are simulated; the proportion of repetitions where an event occurs estimates its probability.​

Law of large numbers in simulation

As the number of simulated repetitions increases, the proportion of times the event occurs will get closer to its true probability.​

Independent random phenomena - definition

Two random phenomena are independent if knowing the outcome of one does not change the probabilities for the outcomes of the other.​

Independence in coin tossing

Repeated tosses of a fair coin are plausibly independent: the coin has "no memory," so the result of one toss does not affect the next.​

Simulation step 1 - give a probability model

Specify the possible outcomes and their probabilities (for example, coin toss: P(H) = 0.5, P(T) = 0.5, tosses independent).​

Simulation step 2 - assign digits

Assign random digits (or pairs of digits) to represent outcomes in a way that matches the probabilities in the model.​

Simulation step 3 - simulate many repetitions

Use random digits to generate many trials, then record whether the event of interest occurs in each repetition.​

Example - coin run of three heads or tails

To estimate the probability of at least three consecutive heads or tails in 10 tosses, simulate many 10‑toss sequences using random digits and check for such runs.​

Assigning digits - equal probabilities example

For a fair coin, one digit can simulate a toss by letting odd digits represent heads and even digits represent tails.​

Assigning digits - unequal probabilities example

To simulate a group where 40% are age 40+ and 60% are under 40, assign 0-3 to "age 40+" and 4-9 to "under 40."​

Assigning digits - three outcomes example

For 30% age 40+ no plans to retire, 10% age 40+ planning to retire soon, 60% under 40, use digits 0-2, 3, and 4-9 to represent the three categories.​

Checking independence with proportions

For coin tosses, independence implies that the proportion of times a toss is followed by the same outcome (HH or TT) should be close to 0.5 over many tosses.​

Checking independence with correlation

If two random numeric outcomes are independent, their correlation should be close to 0; a strong straight‑line pattern suggests lack of independence.​

Scatterplots and independence

If two phenomena are independent, a scatterplot of paired outcomes should show no overall pattern; visible structure suggests dependence.​

More elaborate simulations - idea

Complex simulations may involve variable numbers of trials, multiple stages, or probabilities at later stages that depend on earlier outcomes.​

Example - "We want a girl"

A couple plans children until they have a girl or three children; simulate children's sexes using P(girl) = 0.49, P(boy) = 0.51 to estimate the chance of having at least one girl.​

We want a girl - digit assignment

Use pairs of digits 00-48 for girl and 49-99 for boy to simulate each child's sex based on the 0.49/0.51 probabilities.​

We want a girl - variable trial count

For each repetition, read pairs of digits until the couple has either a girl or three children; different repetitions may require different numbers of children.​

Example - kidney transplant stages

Morris faces multiple stages: survive or die in surgery, transplant success or return to dialysis, then survive 5 years or die in each path.​

Kidney transplant - dependence

Probabilities at stage 3 depend on whether the transplant succeeded or dialysis was needed, so later stages are not independent of earlier ones.​

Kidney transplant - tree diagram

A tree diagram organizes the stages and probabilities (surgery outcome, transplant success/dialysis, 5‑year survival) and guides digit assignment for simulation.​

Kidney transplant - result

From long simulations or mathematics, Morris's probability of living at least 5 years is about 0.558.​

Simulation and complex real‑world systems

Simulation is widely used to study queues, climate change, catastrophic failures, and disease spread, where direct experimentation is difficult or impossible.​

Simulation and probability models - connection

Simulation forces careful thinking about the probability model; a good model plus many repetitions yields informative probability estimates.​

Statistics in summary - simulation with random digits

Random digits can simulate random outcomes because each digit has probability 0.1 of being 0-9 and digits in the table are independent.​

Statistics in summary - multi‑stage simulations

To simulate complex phenomena, string together stages, possibly with different probabilities or with lack of independence between stages.​

Statistics in summary - role of tree diagrams

Tree diagrams are helpful tools for displaying multi‑stage probability models and planning simulations based on those models