Welcome to Experimental and Theoretical Probability!
Have you ever flipped a coin to make a decision, checked the weather forecast for rain, or rolled a die in a board game? If so, you have already used probability! In this chapter, we will explore the difference between what should happen in theory and what actually happens when we carry out real experiments.
Don't worry if maths sometimes feels tricky. We will break every single idea down step-by-step with clear examples and everyday situations!
1. The Basics of Probability
Probability measures how likely an event is to happen. We measure it on a scale from \(0\) to \(1\):
• A probability of \(0\) means the event is impossible (for example, rolling an \(8\) on a standard six-sided die).
• A probability of \(0.5\) (or \(\frac{1}{2}\), or \(50\%\)) means an evens chance (like getting Heads on a fair coin).
• A probability of \(1\) means the event is certain (like the sun rising tomorrow).
Important Rule: Probabilities can be written as fractions, decimals, or percentages, but they can never be negative or greater than \(1\).
Key Takeaway: All probabilities lie between \(0\) and \(1\) inclusive: \(0 \le P(A) \le 1\).
2. Theoretical Probability: What Should Happen
Theoretical probability is calculated using maths and logic. It tells us what ought to happen under ideal conditions when all possible outcomes are equally likely (meaning every outcome has the exact same chance of occurring).
The Formula for Theoretical Probability
\(P(\text{Event}) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\)
Step-by-Step Example: Rolling a Fair Die
Suppose you roll a standard, fair six-sided die. What is the theoretical probability of rolling an even number?
Step 1: List all possible outcomes (the sample space): \(\{1, 2, 3, 4, 5, 6\}\).
Total number of possible outcomes = \(6\).
Step 2: Identify the successful outcomes (even numbers): \(\{2, 4, 6\}\).
Number of successful outcomes = \(3\).
Step 3: Put them into our formula:
\(P(\text{Even}) = \frac{3}{6} = \frac{1}{2} = 0.5\)
Did You Know?
The word fair in probability questions simply means unbiased. A fair coin has a \(50\%\) chance of landing on Heads, and a fair die gives every number from \(1\) to \(6\) an equal chance of \(\frac{1}{6}\).
Key Takeaway: Theoretical probability is based purely on theory and symmetry without needing to run an actual experiment.
3. Experimental Probability (Relative Frequency): What Actually Happens
What if you want to know the probability of a drawing pin landing point-up, or the chance of a dropped piece of toast landing butter-side down? You cannot calculate this theoretically because the shapes are not symmetrical. Instead, you have to perform an experiment (or trial) and collect real data!
Experimental probability (also called relative frequency) is the probability calculated from the results of an experiment or sample.
The Formula for Relative Frequency
\(\text{Relative Frequency} = \frac{\text{Frequency of the event}}{\text{Total number of trials}}\)
Step-by-Step Example: Spinning a Spinner
Chloe spins a coloured spinner \(50\) times. It lands on Blue \(14\) times.
Step 1: Identify the number of times the event occurred: \(\text{Frequency} = 14\).
Step 2: Identify the total number of trials: \(\text{Total trials} = 50\).
Step 3: Calculate relative frequency:
\(\text{Relative Frequency of Blue} = \frac{14}{50} = \frac{7}{25} = 0.28\)
Key Takeaway: Experimental probability is calculated from real experimental data: \(\text{Relative Frequency} = \frac{\text{How many times it happened}}{\text{How many times you tried}}\).
4. The Law of Large Numbers: Connecting the Two
Why doesn't a coin always land on Heads exactly \(5\) times out of \(10\) flips? Because in a small number of trials, chance and randomness cause short-term variation.
However, as you perform more and more trials, something amazing happens: the experimental probability gets closer and closer to the true theoretical probability. This fundamental concept is known as the Law of Large Numbers.
Comparing Small vs Large Samples
• Small number of trials (e.g., \(10\) coin flips): You might get \(7\) Heads (\(\text{relative frequency} = 0.7\)). This is normal variation.
• Large number of trials (e.g., \(1,000\) coin flips): You might get \(503\) Heads (\(\text{relative frequency} = 0.503\)). This is extremely close to the theoretical probability of \(0.5\).
Memory Tip
Think of it as "More Trials = More Reliable". If you want the most accurate estimate of a probability, always use the experiment with the largest number of trials!
Key Takeaway: Increasing the number of trials reduces experimental error and gives a better, more reliable estimate of the true probability.
5. Expected Frequency: Predicting Future Results
Once you know the probability of an event (either theoretical or experimental), you can predict how many times that event will happen over a given number of future trials. This is called the expected frequency.
The Expected Frequency Formula
\(\text{Expected Frequency} = n \times P(\text{Event})\)
Where \(n\) is the total number of future trials and \(P(\text{Event})\) is the probability of the event.
Worked Example 1: Using Theoretical Probability
A fair standard die is rolled \(300\) times. How many times would you expect to roll a \(5\)?
• The theoretical probability of rolling a \(5\) is \(P(5) = \frac{1}{6}\).
• Total trials \(n = 300\).
• \(\text{Expected Frequency} = 300 \times \frac{1}{6} = 50\).
You would expect to roll a \(5\) exactly \(50\) times.
Worked Example 2: Using Experimental Probability
A factory tests light bulbs. From previous testing, the relative frequency of a bulb being faulty is \(0.02\). If the factory produces \(4,500\) bulbs in a day, how many are expected to be faulty?
• \(P(\text{Faulty}) = 0.02\)
• Total trials \(n = 4500\)
• \(\text{Expected Frequency} = 4500 \times 0.02 = 90\)
The factory should expect \(90\) faulty bulbs.
Key Takeaway: To find expected frequency, simply multiply the number of trials by the probability.
6. Checking for Fairness and Bias
We can use experimental probability to test whether a gaming tool (like a coin, die, or spinner) is fair or biased (unfair).
How to Detect Bias
1. Find the theoretical probability assuming the tool is fair.
2. Conduct a large number of trials and calculate the relative frequency.
3. Compare the two values: If the relative frequency is significantly different after hundreds of trials, the tool is likely biased!
Example: Is this die biased?
A student rolls a die \(600\) times and gets the number \(6\) a total of \(210\) times.
• Expected frequency for a fair die: \(600 \times \frac{1}{6} = 100\) times.
• Observed frequency: \(210\) times.
• Relative frequency: \(\frac{210}{600} = 0.35\) (compared to theoretical \(\frac{1}{6} \approx 0.167\)).
Because \(210\) is much higher than the expected \(100\) over a large number of trials, there is strong evidence that the die is biased towards \(6\).
Key Takeaway: A result that differs vastly from expectation in a large sample suggests bias, not just random luck.
7. Common Mistakes to Avoid
Mistake 1: The "Gambler's Fallacy"
Thinking that past results change future independent events. If you flip a fair coin and get Tails \(5\) times in a row, the probability of getting Heads on the next flip is still \(0.5\). The coin has no memory!
Mistake 2: Writing probabilities greater than \(1\)
If your calculation gives an answer like \(1.4\) or \(\frac{5}{4}\), stop and check your working. A probability can never exceed \(1\).
Mistake 3: Confusing frequency and relative frequency
Frequency is a whole count (e.g., \(15\) times). Relative frequency is a fraction or decimal between \(0\) and \(1\) (e.g., \(\frac{15}{50} = 0.3\)).
8. Quick Summary Review
• Theoretical Probability: Based on mathematical reasoning: \(\frac{\text{Successful outcomes}}{\text{Total outcomes}}\).
• Experimental Probability (Relative Frequency): Based on collected data: \(\frac{\text{Frequency of event}}{\text{Total trials}}\).
• Large Samples: As the number of trials increases, relative frequency gets closer to theoretical probability.
• Expected Frequency: Calculate expected occurrences using \(n \times P(\text{Event})\).