Counting and Probability: Conducting Simulations

Example: Simulations and Probability

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$$\text{Probability of any outcome of a chance process is between 0 and 1.}$$ $$\text{ It describes the proportion of times the outcome would occur}$$ $$\text{in a series of repetitions. }$$

We can calculate and predict the probability of different events, that is called the theoretical probability.

We can also conduct trials to find the probability, this is called the experimental probability.

What Simulations? $\\$

It is a lot of work and sometimes very hard to conduct trials to find the experimental probability––especially if we need to conduct a large number of trials. Conducting simulations allow us to find the experimental probability without all the trouble.

Coin Toss Simulation $\\$

Let's toss a fair coin (head and tail are equally likely to show up) and calculate the probability of getting a head.

Generally, the more trials we run, the more accurate the probability.

Let's start with tossing a fair coin $10$ times with an online simulator.

Let's toss the coin $110$ times.

Let's toss the coin $1000$ times.

What is a simulation and why is it helpful?

A simulation is a way to model or imitate events in such a way that the outcome is representative of a real world result. In other words, it is a way to learn about the possible outcomes of a situation without the actual situation occurring. Simulations are conducted when it is too difficult, dangerous, time-consuming, or costly to conduct an experiment. The results of the simulation can be thought of as a hypothesis as to what would actually happen.

How to conduct a simulation?

$1.$ When conducting a simulation begin by finding the possible outcomes.

Let's say we want to know the probability of guessing and getting three questions correct out of five. In this case there are two possible outcomes per question, correct and incorrect.

$2.$ Once the possible outcomes have been identified, decide on a simulation method that fits the scenario and assign each possible real world outcome to a possible outcome in the simulation.

Let's continue with the above scenario. If we want to know the probability of guessing and getting three out of five questions correct we can flip a coin. We can assign $\color{red}{\text{heads}}$ to a $\color{red}{\text{correct}}$ answer and $\color{blue}{\text{tails}}$ to an $\color{blue}{\text{incorrect}}$ answer.

Note: A coin flip simulation can be conducted anytime there are two possible outcomes.

$3.$ Next, begin conducting the situation, completing several trials.

In the above situation, we want to know the probability of getting three questions correct out of five. This means we should flip the coin five times, once per question, per trial.

Note: It is important to conduct multiple trials in order to get an end result that is accurate. The more trials conducted, the closer we get to an accurate theoretical probability.

$4$. Finally, analyze the simulated outcome and report your findings.

The table below represents the results of the simulation described throughout this section.

In this simulation ten trials were conducted with five coin tosses per trial. In the case a coin landed on heads at least three times in a trial, the trial is consider a "success" (the successful trials represent the event that some one does guess three out of five questions correctly). The results can then be analyzed to give an experimental probability that someone will guess three out of five questions correctly.

$$P_{experimental}=\frac{\text{success}}{\text{total trials}}=\frac{5}{10}=0.5$$

(Video) Simulations

by MathHelp.com

Math Help.com has a good simple video explaining what simulations are and how to conduct them.

$\large{\text{Summary}}$

Now, here's an example problem that gives a sample simulation scenario (say that three times fast).