Solving Probability Problems Using Simulation (VCE SSCE Mathematical Methods): Revision Notes

Solving Probability Problems Using Simulation

What is simulation?

Simulation is a practical technique that helps us estimate probabilities when problems are too difficult to solve using direct mathematical methods. By running repeated random trials and recording the results, we can find approximate answers to complex probability questions.

Some probability problems cannot be solved easily with formulas alone. Simulation provides a way to get useful estimates in these situations by:

• Creating a model that represents the real situation
• Running the model many times
• Analysing the results to estimate the probability

An example of using simulation

Let's work through a complete example to understand how simulation works in practice.

The problem

What is the probability that a family with six children will have at least four girls?

While there are mathematical methods to solve this exactly, we can estimate the answer using simulation instead.

Making assumptions

Before we can simulate, we need to make clear assumptions about the situation. For this problem, we assume:

• The probability of each child being female is $0.5$
• The sex of each child is independent of the sex of the other children (meaning the probability of a female child is always $0.5$, regardless of previous children)

Setting up the simulation model

Since the probability of a female child is $0.5$, we can use a fair coin as our simulation tool:

• Heads represents a female child
• Tails represents a male child

One trial consists of tossing the coin six times (representing one family with six children). The result of each trial is the number of heads obtained.

Conducting trials

Here's what five example trials might look like:

Trial number	Simulation results	Number of heads
1	T H H T T H	3
2	H H H T H T	4
3	H H H T H H	5
4	H T T T H T	2
5	H T H H H H	5

To get a reliable estimate, we need to conduct many more trials. A good rule is to perform at least 50 trials.

Results from 50 trials

After completing 50 trials of tossing the coin six times, we might obtain these results:

This table shows us that:

• 0 heads occurred 1 time
• 1 head occurred 5 times
• 2 heads occurred 12 times
• 3 heads occurred 15 times
• 4 heads occurred 13 times
• 5 heads occurred 3 times
• 6 heads occurred 1 time

Interpreting the results

We wanted to find the probability of getting at least four girls (at least four heads). Looking at our results:

• 4 heads: 13 times
• 5 heads: 3 times
• 6 heads: 1 time
• Total: $13 + 3 + 1 = 17$ times

Therefore, we estimate the probability as:

$$\text{Probability} = \frac{17}{50} = 0.34$$

Further examples of using simulation

Simulation can also estimate other quantities of interest, such as average values, maximum values, or the number of attempts needed to achieve a certain outcome.

Using technology to generate random numbers

While coins and dice work well for simple simulations, many situations require more flexibility. For example, if we needed to simulate collecting eight different souvenirs rather than six, we would need to generate random integers from 1 to 8.

Technology allows us to:

• Generate any range of random numbers
• Conduct many more trials quickly
• Handle more complex simulations

Here's how to use calculators to generate random numbers:

Using the TI-Nspire calculator

Setting the random seed:

1. In a Calculator page, go to menu > Probability > Random > Seed
2. Enter the last 4 digits of your phone number (or any 4-digit number)
3. This ensures your random numbers differ from the calculator default

Generating random numbers between 0 and 1:

• Use menu > Probability > Random > Number
• The command rand() generates a random decimal between 0 and 1

Generating random integers:

• Use menu > Probability > Random > Integer
• For example, randInt(2, 4, 5) generates five random integers between 2 and 4 inclusive
• The format is: randInt(minimum, maximum, quantity)

Using the Casio ClassPad calculator

Accessing the catalog:

1. In Main, press the Keyboard button
2. Tap the symbol at the bottom of the left sidebar
3. Select Catalog and scroll to the letter R

Generating random numbers between 0 and 1:

• Select rand() from the catalog
• Tap EXE

Generating random integers:

• Select rand() from the catalog
• Type the minimum and maximum values (e.g., rand(1, 6))
• Tap EXE multiple times to generate multiple random integers

Generating lists of random numbers:

• For random decimals: Select randList(), type the quantity (e.g., 10), tap EXE
• For random integers: Select randList(), type quantity, minimum, and maximum (e.g., randList(20, 1, 30)), tap EXE

Remember!