Probability
Experiments and simulations
You cannot flip a real cat to find out whether a kitten will be male or female. But you can flip a coin — and if the chances match, the coin tells you what you need to know.
What a simulation is
A SIMULATION uses something easy to do — like flipping a coin or rolling a dice — to stand in for something you cannot easily test. One way of using simulations is to program a computer to use data from previous outcomes, then work out what would happen if something were done thousands of times. Simulations like that can predict things such as whether a goal is likely to be scored from different positions on a football pitch.
When is a simulation appropriate?
A simulation only works if the chances MATCH the real thing. To check, describe the likelihood of both: 1. What are the real outcomes, and how likely is each? 2. What are the outcomes of your stand-in, and how likely is each? If the two lists match, the simulation is appropriate. If they do not, it will give you the wrong answer — however many times you run it.
The kitten example
Marcus wants to simulate whether a cat will have male or female kittens. He flips a coin: heads means male, tails means female. THE REAL OUTCOMES: a kitten could be male or female. Male is an even chance; female is an even chance. The two are equally likely. THE COIN: it could land heads or tails. Heads is an even chance; tails is an even chance. The two are equally likely. The two lists match, so YES — a flipped coin is an appropriate way to simulate this.
A simulation that does NOT work
Rachel programs a spreadsheet to simulate 50 coin flips. It creates random numbers from 1 to 3. She says odd numbers mean heads and even numbers mean tails. That is wrong. Among 1, 2 and 3 there are TWO odd numbers (1 and 3) but only ONE even number (2). So her 'heads' is twice as likely as her 'tails' — while a real coin is an even chance either way. The chances do not match, so the simulation is not appropriate. Using 1 to 2, or 1 to 4, would fix it.
Running an experiment
An EXPERIMENT is where you actually do it and record what happens. Roll two dice and find the difference between the numbers. Record each result in a tally chart, then count the FREQUENCY — how many times each outcome happened. A bar chart makes the pattern easy to see. Here is the important part: those outcomes are NOT all equally likely. A difference of 0 needs both dice to match, but a difference of 1 can happen many different ways. If your results look surprising, do MORE trials. A small number of rolls can easily mislead you; a large number gets closer to the truth.
Worked examples
Is a coin a good simulation for a kitten's sex?
- The real outcomes: a kitten is male or female — equally likely.
- The coin: heads or tails — equally likely.
- Compare the two lists. Both have two equally likely outcomes.
Answer: Yes — the chances match, so the coin is an appropriate simulation.
Why 'random 1 to 3' fails for a coin
- Odd numbers are 1 and 3 — that is two outcomes.
- Even numbers are just 2 — that is one outcome.
- So odd is twice as likely as even, but a real coin is an even chance.
Answer: Not appropriate — the chances do not match.
Simulating a dice with a spinner
- A dice has six outcomes, each equally likely.
- A fair spinner with six equal sections also has six equally likely outcomes.
- The two lists match.
Answer: Yes — a six-section fair spinner can simulate a dice.
Recording an experiment
- Roll two dice 30 times and find the difference each time.
- Put a tally mark next to that difference in your table.
- Count the tallies to get the frequency for each outcome.
Answer: The frequencies show which differences came up most often.
Where you see this in real life
- Weather forecasters run simulations thousands of times to say there is a '70% chance of rain'.
- Flipping a coin to decide who goes first is fair because both players have exactly the same chance.
- A game designer tests a new dice game by simulating it on a computer before printing it.
Key words
- Simulation
- Using something easy to do to stand in for something you cannot easily test.
- Experiment
- Actually doing something and recording what happens.
- Outcome
- One of the things that could happen.
- Equally likely
- Having exactly the same chance of happening.
- Frequency
- How many times an outcome happened.
- Tally chart
- A table where you make a mark each time an outcome happens.
- Trial
- One go of an experiment, such as a single roll of the dice.