Simulations
Simulations are used for finding probabilities for compound events. The probability model of a random phenomenon consists of a sample space of possible outcomes, associated events, random variables, and a probability measure that specifies probabilities of events and determines distributions of random variables.
Simulation in detail
Simulation involves employing a probability model to artificially recreate a random phenomenon, usually by the aid of a computer. Given a probability model, we will simulate outcomes, occurrences of events, and values of random variables, in accordance with the specifications of the probability measure. Probabilities may be interpreted as long term relative frequencies. As a result, the probability of an occurrence will be approximated by simulating, in keeping with the assumptions of the probability model. The random phenomenon an oversized number of times and computing the ratio of repetitions on which the event occurs. Simulation can be used to approximate probabilities of events, distributions of random variables, future averages, and other characteristics.
In general, a simulation involves the following subsequent steps:
1. Set up. Define a probability space, as well as the related random variables and events. The probability measure encodes all the assumptions of the model, but the probability measure is usually specified indirectly only. It may be discovered in as simple a way as FLIPPING A GOOD COIN TEN TIMES AND COUNTING THE NUMBER OF HEADS. However, even when the setup can be described simply, translating the setup into code may be challenging.
2. Simulate. Simulate in keeping with the probability measure—outcomes, occurrences of events, and values of random variables.
3. Summarize. Summarize simulation output in plots and summary statistics (relative frequencies, means, standard deviations, correlations, etc,) to explain and approximate probabilities, distributions, and related characteristics.
You might ask: if we have access to the probability measure, then why do we need simulation to approximate probabilities? Can’t we just compute them?
Remember that the probability measure is usually specified indirectly. The probability measure represents the underlying assumptions under which probabilities of events of interest are determined. But in most situations, the probability measure doesn’t provide a precise formula for computing the probability of any particular event. And in many cases, it’s impossible to enumerate all possible outcomes.
For example,
A probabilistic model of a selected Atlantic hurricane doesn’t provide a mathematical formula for computing the probability that the hurricane will make landfall within the U.S. Nor does the model provide a comprehensive list of the uncountability of the many possible paths of the hurricane. Rather, the model reflects a collection of assumptions under which possible paths are often simulated to approximate probabilities of events of interest.
Simulation could be a powerful tool for investigating probability models and solving complex problems.
How to simulate experiments involving equally likely outcomes?
When all possible outcomes of an action have precisely the same probability, we can say that the action has equally likely outcomes. When flipping a coin, heads or tails are equally likely (each includes a probability of \(\frac{1}{2}\)). When a normal die is rolled, the six possible outcomes are equally likely (each features a probability of \(\frac{1}{6}\)).
The act of rolling a pair of dice leads to 36 equally likely outcomes (using the elemental counting principle: 6 x 6 = 36). Although these 36 outcomes are equally likely, the chances for various dot sums don’t seem to be easy since a dot sum is an incident and therefore, the number of successful outcomes varies with the dot sum.
Example simulating outcomes that are equally likely
Example 1: A dog has three puppies. The gender of every puppy is equally likely.
Design a simulation involving 20 trials that you can use to model the genders of the puppies.
Choose an experiment that has two equally likely outcomes for every event (gender), like flipping three coins. Let heads (H) represent a male and tails (T) represent a female.
Solution:
Use your simulation to find the experimental probability that each of the three puppies is male.
To find the experimental probability, perform 20 trials of the simulation. The table shows the results. Find the number of outcomes that represent 3 females, TTT.
HTH | HTT | HTT | HTH | HTT |
|---|---|---|---|---|
TTT | HTT | HHH | TTT | HTT |
HTH | HTT | HHH | HTH | HTT |
HTT | HTH | TTT | HTT | HTH |
TTT occurred 3 times.
There are an overall of 20 trails.
P ( 3 females) \(=\frac{3}{20}\)
The experimental probability is \(\frac{3}{20}\), 0.15, 15%.
How to simulate experiments involving non equally likely outcomes?
Let’s understand this with an example below:
You have a 60% chance of winning a parlor game and a 20% chance of winning a cardboard game. Design and use a simulation involving 50 randomly generated numbers to estimate the probability of winning both games.
Solution:
Use a simulation with randomly generated numbers from 0 to 99. Let the digits 1 through 6 within the tens place represent winning the parlor game. Let the digits 1 and a pair within the ones place represent winning the cardboard game.
Use the random number generator on a graphing calculator to come up with the numbers. The table shows the results. Find the number of outcomes that represent winning both games.
52 | 66 | 73 | 68 | 75 | 28 | 35 | 47 | 48 | 02 |
|---|---|---|---|---|---|---|---|---|---|
16 | 68 | 49 | 03 | 77 | 35 | 92 | 78 | 06 | 06 |
58 | 18 | 89 | 39 | 24 | 80 | 32 | 41 | 77 | 21 |
32 | 40 | 96 | 59 | 86 | 01 | 12 | 00 | 94 | 73 |
40 | 71 | 28 | 61 | 01 | 24 | 37 | 25 | 03 | 25 |
P (Win both games) \(=\frac{7}{50}\) ( the number 7 meets the factors and there are a total number of fifty trials).
Therefore, the experimental probability is \(\frac{7}{50}\), 0.14, or 14%.
A probability model of a random phenomenon consists of a sample space of possible outcomes, associated events, random variables, and a probability measure that specifies probabilities of events and determines distributions of random variables.
From the probability model, we will simulate outcomes, occurrences of events, and values of random variables, in step with the specifications of the probability measure.
Simulation may be used for approximating probabilities of events, distributions of random variables, long term averages, and other characteristics.
Simulation could be a powerful tool for investigating probability models and solving complex problems.
When all the possible outcomes of an action have precisely the same probability, we can understand that the action has equally likely outcomes.
There are three styles of simulations: live, virtual, and constructive.
First, a setup is required to define a probability space, related random variables and events. The probability measure encodes all the assumptions of the model. However, the probability measure is usually specified indirectly only. Then, Simulate in step with the probability measure outcomes, occurrences of events, and values of random variables. Finally, summarize the simulation output in plots and summary statistics.
The steps for conducting simulation by using a computer are: Defining the system, Building the probability model, Conducting the experiments, Analyzing the output and reporting the leads to the personal computer.