Experiments and simulations
🎯 In this topic you will
- Carry out experiments and simulations to investigate probability.
🧠 Key Words
- simulation
Show Definitions
- simulation: A model or digital process used to imitate a real experiment so that outcomes can be studied without performing the actual physical trial.
Understanding Simulations
S imulations can help us to know the likelihood of an outcome happening.
Using Computers for Simulations
O ne way of using simulations is to program a computer to use data from previous outcomes to work out what would happen if something was done thousands of times.
Predicting Real-World Events
T hese simulations can try to predict things like whether a goal is likely to be scored from different positions on a football pitch.
❓ EXERCISES
1. You are going to do an experiment to investigate the likelihood of different outcomes when you roll two dice and find the difference between the two numbers.
a. What different outcomes can you get if you roll two dice and find the difference between the two numbers?
b. Do you think that all the outcomes are equally likely?
c. Copy the table and complete the first column.
| The difference between the two numbers | Tally | Frequency |
|---|---|---|
d. Roll two dice $30$ times and record the results in your table.
e. Draw a bar chart to show your results.
f. Do your results suggest that the outcomes are all equally likely?
g. What could you do to check?
👀 Show answer
a. The possible differences when you roll two dice and take the difference between the numbers are $0, 1, 2, 3, 4$ and $5$.
b. No. The outcomes are not all equally likely. Differences of $1$ and $2$ happen more often than $0$ or $5$ when you look at all the possible pairs of dice scores.
c. The first column of the table should list the differences $0, 1, 2, 3, 4, 5$ (one in each row).
d. Your tallies and frequencies should show $30$ results in total (so the frequencies add up to $30$). The exact numbers will vary from experiment to experiment.
e. A correct bar chart has the differences $0$ to $5$ along the horizontal axis and the frequencies from your table on the vertical axis, with bars drawn to the right heights.
f. Typically the results do not suggest that all outcomes are equally likely; some differences (such as $1$ or $2$) usually appear more often than others.
g. To check, you could repeat the experiment many more than $30$ times (or combine results with your classmates, or use a computer simulation) and see whether the pattern of frequencies is consistent.
2. There are five shapes in a bag. Arun took a shape out of the bag, looked at it and put it back into the bag. He did this eight times. These are the shapes he saw.
a. Write true or false for each statement.
i. There was at least one sphere in the bag.
ii. There was definitely not a triangular prism in the bag.
iii. There might be more than one cube in the bag.
iv. There is definitely only one cylinder in the bag.
b. Sketch the five shapes that you predict might be in the bag.
c. Explain why you chose those five shapes for your prediction.
d. What could you do to improve your prediction without looking into the bag?
👀 Show answer
a i.True. At least one sphere was definitely in the bag because Arun saw a sphere during his draws.
a ii.False. We did not see a triangular prism, but that does not prove there was definitely not one in the bag; it might simply not have been picked in the eight draws.
a iii.True. Arun saw several cubes, so it is possible that there was more than one cube in the bag.
a iv.False. Seeing one cylinder does not prove there is only one cylinder in the bag; there could be more cylinders that just were not selected.
b. One reasonable prediction is that the bag contains two cubes, one sphere, one cylinder and one triangular-based pyramid (five shapes altogether). Other similar combinations that match the observations are also acceptable.
c. This prediction uses the fact that cubes were seen most often, so there may be more than one cube, while the sphere, cylinder and triangular-based pyramid were each seen at least once, so it is sensible to include each of them in the bag.
d. To improve the prediction without looking into the bag, Arun could repeat the experiment many more times, carefully recording each shape he draws, and then use these results to refine his estimate of which shapes, and how many of each, are in the bag.
3. Rachel programmed a spreadsheet to simulate $50$ coin flips. The spreadsheet creates random numbers from $1$ to $3$.
Rachel says, ‘The odd numbers represent coin flips that land on heads. The even numbers represent coin flips that land on tails.’
Rachel’s spreadsheet list looks like this:
$3\ 1\ 2\ 2\ 2\ 2\ 3\ 3\ 1\ 3$
$1\ 3\ 2\ 3\ 1\ 3\ 3\ 3\ 1\ 2$
$3\ 1\ 3\ 2\ 3\ 2\ 3\ 3\ 2\ 1$
$3\ 1\ 2\ 3\ 2\ 1\ 1\ 3\ 1\ 2$
$1\ 3\ 2\ 3\ 2\ 2\ 3\ 2\ 2\ 1$
a. How many even numbers are in Rachel’s list?
b. How many odd numbers are in Rachel’s list?
c. Explain what is wrong with Rachel’s simulation.
d. Suggest a better way to simulate the coin flips using random numbers.
👀 Show answer
a. The even numbers in the list are all the $2$s. There are $17$ even numbers.
b. The odd numbers in the list are the $1$s and $3$s. There are $13$ ones and $20$ threes, so $33$ odd numbers altogether.
c. The simulation is not correct because heads and tails are not equally likely. Using numbers from $1$ to $3$, Rachel has two odd values ($1$ and $3$) but only one even value ($2$), so the probability of “heads” is $\dfrac{2}{3}$ and the probability of “tails” is only $\dfrac{1}{3}$ rather than $\dfrac{1}{2}$ each.
d. A better simulation would use random numbers where heads and tails are equally likely, for example random integers from $1$ to $2$ with $1$ = heads and $2$ = tails, or random integers from $1$ to $4$ with $1$ and $2$ = heads and $3$ and $4$ = tails.
4. The weather forecast says that there is an even chance of rain for the next $7$ days.
Sofia made a simulation of the weather using a dice.
Numbers $1, 2$ and $3$ are rainy days.
Numbers $4, 5$ and $6$ are dry days.
Sofia rolled the dice $7$ times to see what might happen. These are her results.
a. How many rainy days are there in Sofia’s simulation?
b. How many dry days are there in Sofia’s simulation?
c. Draw your own table and carry out your own simulation of the weather using Sofia’s rules.
d. How many rainy days are there in your simulation?
e. How many dry days are there in your simulation?
👀 Show answer
From the diagram, the dice show (from Day $1$ to Day $7$): $5, 4, 1, 1, 3, 3, 2$.
a. Rainy days are when the dice shows $1, 2$ or $3$. There are $5$ rainy days (Days $3, 4, 5, 6$ and $7$).
b. Dry days are when the dice shows $4, 5$ or $6$. There are $2$ dry days (Days $1$ and $2$).
c. Your own simulation should follow the same rules, rolling a fair dice $7$ times, recording each result and then classifying each day as rainy or dry.
d. The number of rainy days in your simulation will depend on your own dice results; any value from $0$ to $7$ is possible.
e. The number of dry days in your simulation will also depend on your own results; it should satisfy “rainy days $+$ dry days $= 7$”.
🧠 Think like a Mathematician
Use computer software to generate 100 random numbers from 1 to 20.
a. How many times does the number 1 appear in the list?
b. How many numbers in the list are greater than 10?
c. How many numbers in the list are less than 6?
d. Generate another set of 100 random numbers from 1 to 20. Answer questions a, b and c again using the 100 new numbers.
e. Zara says that the numbers 1 to 20 are all equally likely, so each number will appear the same number of times in the list.
Explain why Zara is wrong.
Write your explanation clearly. Then reread what you have written and check:
- Did you write about the results of your simulation?
- Did you explain how, as the computer produces more and more random numbers, the results become closer to showing that the numbers are all equally likely?
- You are characterising when you describe the sets of random numbers.
- You are convincing when you explain why Zara is wrong.
👀 show answer
- a. The exact answer depends on your random list. Because each number from 1 to 20 has probability $\dfrac{1}{20}$, you would expect the number 1 to appear about $\dfrac{1}{20} \times 100 = 5$ times, but the actual count might be a bit more or less.
- b. Numbers greater than 10 are 11 to 20, which is 10 numbers. Each has probability $\dfrac{1}{20}$, so you would expect about $\dfrac{10}{20} \times 100 = 50$ of the 100 values to be greater than 10. Your experimental result should be close to, but not exactly, 50.
- c. Numbers less than 6 are 1 to 5, which is 5 numbers. You would expect about $\dfrac{5}{20} \times 100 = 25$ values to be less than 6, again with some random variation.
- d. In the second set of 100 random numbers the answers should again be roughly 5 occurrences of each individual number, about 50 numbers greater than 10, and about 25 numbers less than 6. Comparing the two experiments should show that the totals are not identical, but are generally close to these expected values.
- e. Zara is wrong because “equally likely” does not mean “appears exactly the same number of times” in a finite experiment. In 100 random numbers from 1 to 20, each number has probability $\dfrac{1}{20}$, but chance variation means some numbers will appear more often and some less often. As you generate larger and larger lists (for example 1000 or 10 000 numbers), the proportion of each number tends to get closer to $\dfrac{1}{20}$, but the actual counts will almost never be exactly equal.
- Characterising. A strong description mentions which numbers occur more or less often, quotes approximate frequencies or proportions, and compares them with the expected probabilities.
- Convincing. A convincing explanation of why Zara is wrong uses both the simulation results and probability reasoning: it refers to the observed data and explains that randomness causes variation, even when outcomes are equally likely.
💡 Quick Math Tip
More trials give better probability estimates: When you use a computer to generate many random numbers, the more numbers you produce, the closer the results will get to the true probabilities, even though the counts for each number will not be exactly the same.