Empirical Samples

Feature	Small Sample Size (e.g., 10 trials)	Large Sample Size (e.g., 1,000 trials)
Reliability	Unreliable estimate of probability	More reliable estimate of probability
Fluctuation	Highly susceptible to random chance	Smooths out short-term variations
Convergence	Far from theoretical probability	Very close to theoretical probability

Finding the Best Estimate

• If three different students do the same coin-flipping experiment, whose result should you trust?
• In an exam, you will often need to decide which experiment gives the "best estimate" of a probability. The most reliable estimate always comes from the experiment with the largest number of trials.
• When multiple people conduct the same experiment, you do not average their probabilities. Instead, you find the best estimate by combining all their results, totaling the frequencies and the total trial numbers.

Stuart flips a coin 10 times and gets 7 Tails. Maxine flips the same coin 50 times and gets 30 Tails. Calculate the best estimate for the probability of getting Tails.

Step 1: Identify the total number of successful outcomes.

• $\text{Total Tails} = 7 + 30 = 37$

Step 2: Identify the total number of trials across all experiments.

• $\text{Total Trials} = 10 + 50 = 60$

Step 3: Calculate the best estimate using the combined totals.

• $\text{Best Estimate} = \frac{37}{60}$

Avoiding Bias in Experiments

• Imagine trying to find the most popular sport in a school by only asking the football team.
• For an empirical distribution (the set of observed experimental results) to converge on a theoretical probability distribution, the sample must be an unbiased sample. This means the experiment must be entirely random, every member of the population must have an equal chance of selection, and each trial must be independent.
• A sample suffering from bias does not accurately represent the population and does not give every outcome an equal chance of occurring. To ensure fairness, objects must be replaced between trials to keep probabilities constant.
• Valid methods to ensure randomness include using a random number generator or picking names from a hat. In stratified sampling, the sample reflects the proportions of different groups in the population.

$$\text{Sample Size for Stratum} = \frac{\text{Stratum Population}}{\text{Total Population}} \times \text{Total Sample Size}$$

Using Expected Frequency to Predict Outcomes

• Factories test a small batch of products to predict how many faulty items might be in an order of a million.
• Once you have calculated a probability, you can use it to estimate future outcomes. The expected frequency is the number of times you estimate an outcome will happen in a given number of trials.
• It is calculated by multiplying the probability (theoretical or experimental) by the total number of future trials.

$$\text{Expected Frequency} = \text{Probability} \times \text{Total number of trials}$$

A biased 4-sided spinner has a relative frequency of $0.35$ for landing on 'Red'. If the spinner is spun $400$ times, estimate the number of times it will land on Red.

Step 1: State the expected frequency formula.

• $\text{Expected Frequency} = \text{Probability} \times \text{Total number of trials}$

Step 2: Substitute the known values into the equation.

• $\text{Expected Frequency} = 0.35 \times 400$

Step 3: Calculate the final estimate.

• $\text{Expected Frequency} = 140$ times